Breaking news - Coin Tosses are not 50-50 (1 Viewer)

[FUBAR]

Just came back from a party - 25 people in the room - all very successful, college educated. One person a Doctor, another a CPA, and one person who worked in the West Wing of the White House as an intern two summers ago - writing portions of Obama's speeches. Not one single person knew the answer. Common knowledge indeed. I think we all know who's full of #### here.

I'd be willing to bet the number who cared to be fewer than the number who knew the answer.

If I'm wrong, I'll just double the next bet.

 

[cstu]

This is common knowledge. No way you asked anyone prepping for the P1, let alone an actual actuary that missed that.

I've heard this before but had forgotten it. Treating it as 'common knowledge' among non- is

 

[rockaction]

Two people are flying in a hot air balloon and realize they are lost. They see a man on the ground, so they navigate the balloon to where they can speak to him. They yell to him, “Can you help us – we’re lost.” The man on the ground replies, “You’re in a hot air balloon, about two hundred feet off the ground.”

One of the people in the balloon replies to the man on the ground, “You must be an actuary. You gave us information that is accurate, but completely useless.” The actuary on the ground yells to the people in the balloon, “you must be in marketing.”

They yell back, “yes, how did you know?” The actuary says,” well, you’re in the same situation you were in before you talked to me, but now it’s my fault.”

Riposte:

“A biologist, a chemist, and a statistician are out hunting. The biologist shoots at a deer and misses 5ft to the left, the chemist takes a shot and misses 5ft to the right, and the statistician yells, ‘We got ‘im!’ ”

 

[Hooper31]

Nerd iFight!

Seriously though, I love teaching statistics. I would much rather teach students to better understand what "random" ** means with regard to stats than algebra. Bothers me a lot that algebra is the focus of almost all high school mathematics instead of statistics.

** I do a specific activity each year with coin flips that usually blows students away. I have them guess at what 100 random flips of a coin will look like, and then I actually have them do the activity. Students generally max out at 4 or 5 consecutive heads or tails when they guess, but in reality almost everyone will have a string of 7, 8, or 9 consecutive heads/tails in their actual data. Humans are pattern machines. We like pattern. We feel more comfortable with the notion of heads and tails taking turns.

 

[Short Corner]

This is common knowledge. No way you asked anyone prepping for the P1, let alone an actual actuary that missed that.

I've heard this before but had forgotten it. Treating it as 'common knowledge' among non- is

I intentionally overstated as his claim that nobody knew the answer in a room full if actuaries is even more

I remember learning it the first time whenI had to take stats with non math types (filled the math grad requirement). It is a very well known prop bet. Being a is not a requirement.

 

[General Tso]

Nerd iFight!

Seriously though, I love teaching statistics. I would much rather teach students to better understand what "random" ** means with regard to stats than algebra. Bothers me a lot that algebra is the focus of almost all high school mathematics instead of statistics.

** I do a specific activity each year with coin flips that usually blows students away. I have them guess at what 100 random flips of a coin will look like, and then I actually have them do the activity. Students generally max out at 4 or 5 consecutive heads or tails when they guess, but in reality almost everyone will have a string of 7, 8, or 9 consecutive heads/tails in their actual data. Humans are pattern machines. We like pattern. We feel more comfortable with the notion of heads and tails taking turns.

The psychology around it is what interests me most. It was really the point of my original post which unfortunately degenerated into a pissing contest about how smart the people in here are. The birthday paradox is defined as a paradox because it so counter-intuitive to normal human thought. In curious as to the evolutionary reasons why the human brain evolved without much capability in this area. Or we can keep trolling each other...

 

[General Tso]

This is common knowledge. No way you asked anyone prepping for the P1, let alone an actual actuary that missed that.

I've heard this before but had forgotten it. Treating it as 'common knowledge' among non- is

I intentionally overstated as his claim that nobody knew the answer in a room full if actuaries is even more I remember learning it the first time whenI had to take stats with non math types (filled the math grad requirement). It is a very well known prop bet. Being a is not a requirement.

Like a typical actuary, you lost the forest through the trees in my post. Second, you used limited information to arrive at a wrong conclusion and then called me a liar. The question I asked was, "how many people do you need to have in a room for the odds to be 50/50 that people have the same birthday?" I also never said how big the room was. It was a smaller setting with about 7 line of business actuaries. Oh, and a calculator was given to the group and I usually give them anywhere from 5-10 minutes. They understood the concept of the probability of pairs, and most were getting at roughly the formula, but not one of them got the answer right. And I wouldn't expect anyone to get that right in that environment. Here's the solution - which ain't exactly easy to calculate, even for an actuary - http://betterexplained.com/articles/understanding-the-birthday-paradox/

 

[Hooper31]

Or we can keep trolling each other...

You're a cotton headed ninny muggins.

 

[xulf]

GT, I am curious as to your background. You come off a certain way in this thread and am legitimately interested in knowing.

 

[DropKick]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

 

[General Tso]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

 

[Dan Lambskin]

Consultant

A shepherd was herding his flock in a remote pasture when suddenly a brand-new BMW advanced out of a dust cloud towards him.

The driver, a young man in an Armani suit, Gucci shoes, Ray Ban sunglasses and YSL tie, leans out the window and asks the shepherd, "If I tell you exactly how many sheep you have in your flock, will you give me one?"

The shepherd looks at the man, obviously a yuppie, then looks at his peacefully grazing flock and calmly answers, "Sure. Why not?"

The yuppie parks his car, whips out his Dell notebook computer, connects it to his AT&T cell phone, surfs to a NASA page on the internet, where he calls up a GPS satellite navigation system to get an exact fix on his location which he then feeds to another NASA satellite that scans the area in an ultra-high-resolution photo. The young man then opens the digital photo in Adobe Photoshop and exports it to an image processing facility in Hamburg, Germany. Within seconds, he receives an email on his Palm Pilot that the image has been processed and the data stored. He then accesses a MS-SQL database through an ODBC connected Excel spreadsheet with hundreds of complex formulas. He uploads all of this data via an email on his Blackberry and, after a few minutes, receives a response.

Finally, he prints out a full-color, 150-page report on his hi-tech, miniaturized HP LaserJet printer and finally turns to the shepherd and says, You have exactly 1,586 sheep.

Thats right. Well, I guess you can take one of my sheep. says the shepherd. He watches the young man select one of the animals and looks on amused as the young man stuffs it into the trunk of his car.

Then the shepherd says to the young man, Hey, if I can tell you exactly what your business is, will you give me back my sheep?

The young man thinks about it for a second and then says, Okay, why not?

Youre a consultant. says the shepherd.

Wow! Thats correct, says the yuppie, but how did you guess that?

No guessing required. answered the shepherd. You showed up here even though nobody called you; you want to get paid for an answer I already knew, to a question I never asked; and you dont know crap about my business..."

"...Now give me back my dog!"

 

[General Tso]

GT, I am curious as to your background. You come off a certain way in this thread and am legitimately interested in knowing.

A lot like you actually.

 

[DropKick]

Consultant

A shepherd was herding his flock in a remote pasture when suddenly a brand-new BMW advanced out of a dust cloud towards him.

The driver, a young man in an Armani suit, Gucci shoes, Ray Ban sunglasses and YSL tie, leans out the window and asks the shepherd, "If I tell you exactly how many sheep you have in your flock, will you give me one?"

The shepherd looks at the man, obviously a yuppie, then looks at his peacefully grazing flock and calmly answers, "Sure. Why not?"

The yuppie parks his car, whips out his Dell notebook computer, connects it to his AT&T cell phone, surfs to a NASA page on the internet, where he calls up a GPS satellite navigation system to get an exact fix on his location which he then feeds to another NASA satellite that scans the area in an ultra-high-resolution photo. The young man then opens the digital photo in Adobe Photoshop and exports it to an image processing facility in Hamburg, Germany. Within seconds, he receives an email on his Palm Pilot that the image has been processed and the data stored. He then accesses a MS-SQL database through an ODBC connected Excel spreadsheet with hundreds of complex formulas. He uploads all of this data via an email on his Blackberry and, after a few minutes, receives a response.

Finally, he prints out a full-color, 150-page report on his hi-tech, miniaturized HP LaserJet printer and finally turns to the shepherd and says, You have exactly 1,586 sheep.

Thats right. Well, I guess you can take one of my sheep. says the shepherd. He watches the young man select one of the animals and looks on amused as the young man stuffs it into the trunk of his car.

Then the shepherd says to the young man, Hey, if I can tell you exactly what your business is, will you give me back my sheep?

The young man thinks about it for a second and then says, Okay, why not?

Youre a consultant. says the shepherd.

Wow! Thats correct, says the yuppie, but how did you guess that?

No guessing required. answered the shepherd. You showed up here even though nobody called you; you want to get paid for an answer I already knew, to a question I never asked; and you dont know crap about my business..."

"...Now give me back my dog!"

A farmer counted his sheep... he had 1,586. But, when he rounded them up, he had 1,600.

 

[DropKick]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

They start out with logic similar to mine

The chance of two people having different birthdays = 1 - 1/365 = 364/365 = .997260

They also state this "sqrt is roughly the number you need to have a 50% chance of a match with n items. sqrt(365) is about 20. This comes into play in cryptography for the birthday attack."

The chance of 20 people having different birthdays = 1 -1/365 - 2/365 - 3/365 - .... - 19/365 = 1-190/365 = 48%

What am I missing?

 

[General Tso]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

They start out with logic similar to mine

The chance of two people having different birthdays = 1 - 1/365 = 364/365 = .997260

They also state this "sqrt is roughly the number you need to have a 50% chance of a match with n items. sqrt(365) is about 20. This comes into play in cryptography for the birthday attack."

The chance of 20 people having different birthdays = 1 -1/365 - 2/365 - 3/365 - .... - 19/365 = 1-190/365 = 48%

What am I missing?

I know just the guy who can help. Oh Short Corner... customer service needed on aisle 2.

 

[cdubz]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

They start out with logic similar to mine

The chance of two people having different birthdays = 1 - 1/365 = 364/365 = .997260

They also state this "sqrt is roughly the number you need to have a 50% chance of a match with n items. sqrt(365) is about 20. This comes into play in cryptography for the birthday attack."

The chance of 20 people having different birthdays = 1 -1/365 - 2/365 - 3/365 - .... - 19/365 = 1-190/365 = 48%

What am I missing?

I am coming up with slightly different probabilities than that site, so maybe I am doing it wrong but still coming up with answer of 23.

Generally probability is multiplicative instead of additive, and in this case re-phrasing the question as what is the probability as what are the odds no one has the same birthday might be helpful.

So odds two people don't have same birthday is 364/365, and odds they do is then 1-(364/365). To expand to 3 people, we add in the 363/365 term as 1-(364/365)*(363/365). That is the 2nd person has

364 unique possibilities whereas the 3rd person only has 363. The final product requires 22 terms (+ 1st person) in order to get over 50%.

Spoiler

1 0.0%

2 0.3%

3 0.8%

4 1.6%

5 2.7%

6 4.0%

7 5.6%

8 7.4%

9 9.5%

10 11.7%

11 14.1%

12 16.7%

13 19.4%

14 22.3%

15 25.3%

16 28.4%

17 31.5%

18 34.7%

19 37.9%

20 41.1%

21 44.4%

22 47.6%

23 50.7%

24 53.8%

25 56.9%

26 59.8%

27 62.7%

28 65.5%

 

[jhib]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

They start out with logic similar to mine

The chance of two people having different birthdays = 1 - 1/365 = 364/365 = .997260

They also state this "sqrt is roughly the number you need to have a 50% chance of a match with n items. sqrt(365) is about 20. This comes into play in cryptography for the birthday attack."

The chance of 20 people having different birthdays = 1 -1/365 - 2/365 - 3/365 - .... - 19/365 = 1-190/365 = 48%

What am I missing?

See Appendix A in that link.

The chance of 20 people having different birthdays is actually = 1 * (1-1/365) * (1-2/365) * (1-3/365) * ... * (1-19/365)

Basically, you need to multiply the chances of two or more things occurring at the same time rather than add (or subtract) them.

ETA: or what cdubz said

 

[grateful zed]

Two people are flying in a hot air balloon and realize they are lost. They see a man on the ground, so they navigate the balloon to where they can speak to him. They yell to him, Can you help us were lost. The man on the ground replies, Youre in a hot air balloon, about two hundred feet off the ground.

One of the people in the balloon replies to the man on the ground, You must be an actuary. You gave us information that is accurate, but completely useless. The actuary on the ground yells to the people in the balloon, you must be in marketing.

They yell back, yes, how did you know? The actuary says, well, youre in the same situation you were in before you talked to me, but now its my fault.

Ha. Pretty good Leroy.An actuary and a farmer were traveling by train. When they passed a flock of sheep in a meadow, the actuary said, There are 1,248 sheep

out there. The farmer replied, Amazing. By chance, I know the owner, and the figure is absolutely correct. How did you count them so

quickly? The actuary answered, Easy, I just counted the number of legs and divided by four.

i doubt his name is leroy.

 

[DropKick]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

They start out with logic similar to mine

The chance of two people having different birthdays = 1 - 1/365 = 364/365 = .997260

They also state this "sqrt is roughly the number you need to have a 50% chance of a match with n items. sqrt(365) is about 20. This comes into play in cryptography for the birthday attack."

The chance of 20 people having different birthdays = 1 -1/365 - 2/365 - 3/365 - .... - 19/365 = 1-190/365 = 48%

What am I missing?

I am coming up with slightly different probabilities than that site, so maybe I am doing it wrong but still coming up with answer of 23.

Generally probability is multiplicative instead of additive, and in this case re-phrasing the question as what is the probability as what are the odds no one has the same birthday might be helpful.

So odds two people don't have same birthday is 364/365, and odds they do is then 1-(364/365). To expand to 3 people, we add in the 363/365 term as 1-(364/365)*(363/365). That is the 2nd person has 364 unique possibilities whereas the 3rd person only has 363. The final product requires 22 terms (+ 1st person) in order to get over 50%.

Spoiler

1 0.0%

2 0.3%

3 0.8%

4 1.6%

5 2.7%

6 4.0%

7 5.6%

8 7.4%

9 9.5%

10 11.7%

11 14.1%

12 16.7%

13 19.4%

14 22.3%

15 25.3%

16 28.4%

17 31.5%

18 34.7%

19 37.9%

20 41.1%

21 44.4%

22 47.6%

23 50.7%

24 53.8%

25 56.9%

26 59.8%

27 62.7%

28 65.5%

Thanks, I can see why my approach was flawed. It was cumulative - in that it tried to capture the possibility of a match on the 2nd (or 3rd, 4th, etc.) but never really adjusted for an earlier match in its "incremental probability". As a result, the numbers are overstated.

I tried your approach in a spreadsheet and agree you both get 23 but the numbers are slightly different. Yours is 49.27 and the exponential method is 49.92. You don't overstate the odds as much as I did but I think it still lacks an adjustment to reflect the match could happen with the 2nd person or the last...

Rounding wouldn't cause that much of a difference. but depending on the computer, this is 364/365 in a calculator and a spreadsheet (32 bit).

0.99726027397260273972602739726027

0.997260273972603

The more I've looked at, the more the exponential "pairs" approach makes sense.

 

[xulf]

GT, I am curious as to your background. You come off a certain way in this thread and am legitimately interested in knowing.

A lot like you actually.

I don't know what you know about me, so I don't know what this means.

 

[renesauz]

Not surprisingly, though one full percentage point is actually a pretty huge variance.

What's more interesting about randomness is the mathematics behind streaks. Most people can't naturally get their heads around it. Let's assume that a coin toss really is a 50/50 event. If you toss the coin 10 times, the odds of getting 7 in a row of heads or tails is somewhere around 2%. If you do it 100 times, the odds go up to 32%.

I got into a debate at work about this in the context of double betting after a loss at the casino. Let's say you are playing blackjack and you are really really good, to the point where each draw is a 50/50 event. My friend was postulating that if you simply double your bet after every loss, you will beat the house. Say you are at a $2 table. You lose your $2 bet, so the next bet you go to $4. If you win, you win back your prior bet and you are $2 ahead. If you lose, the next bet gets raised to $8. Once again, you are betting $8 to win $2. You are basically betting against streaks.

My argument was that at some point - a lot quicker than most people will imagine, you will hit the maximum bet at the table. You'll also be in the ridiculous circumstance of having to bet $256 to win back $2. And the likelihood of 8 straight losses is a LOT higher than people think, particularly when you are at a blackjack table for 4 or 5 hours. If you ever wondered why a casino has maximum bets at the table, this is the reason.

Where math really gets weird is when it comes to probabilities with pairing. For instance, what is the probability that two people have the same birthday. There are 365 possible birthdays, so you might think you need 183 people in a room for there to be a 50/50 chance that two people have the same birthday. In reality, you only need 23. Yup, if you have a room full of 23 people there is a greater than 50/50 chance that two people in the room have the same birthday. I asked this question once to a room full of actuaries and nobody got it right.

All of this helps explain superstition and religion. What people don't realize is that crazy random events are bound to happen to you. It's not God thinking you are special. Just the opposite.

Doing the math in my head, I come up with just 20. The odds of the second person sharing a birthday with the first being 1/365, the third 2/365... and 20 is the sum 1..19/365 ~= 190/365

Please show your work

see link in post 57

They start out with logic similar to mine

The chance of two people having different birthdays = 1 - 1/365 = 364/365 = .997260

They also state this "sqrt is roughly the number you need to have a 50% chance of a match with n items. sqrt(365) is about 20. This comes into play in cryptography for the birthday attack."

The chance of 20 people having different birthdays = 1 -1/365 - 2/365 - 3/365 - .... - 19/365 = 1-190/365 = 48%

What am I missing?

I am coming up with slightly different probabilities than that site, so maybe I am doing it wrong but still coming up with answer of 23.

Generally probability is multiplicative instead of additive, and in this case re-phrasing the question as what is the probability as what are the odds no one has the same birthday might be helpful.

So odds two people don't have same birthday is 364/365, and odds they do is then 1-(364/365). To expand to 3 people, we add in the 363/365 term as 1-(364/365)*(363/365). That is the 2nd person has 364 unique possibilities whereas the 3rd person only has 363. The final product requires 22 terms (+ 1st person) in order to get over 50%.

Spoiler

1 0.0%

2 0.3%

3 0.8%

4 1.6%

5 2.7%

6 4.0%

7 5.6%

8 7.4%

9 9.5%

10 11.7%

11 14.1%

12 16.7%

13 19.4%

14 22.3%

15 25.3%

16 28.4%

17 31.5%

18 34.7%

19 37.9%

20 41.1%

21 44.4%

22 47.6%

23 50.7%

24 53.8%

25 56.9%

26 59.8%

27 62.7%

28 65.5%

Spoiler

Thanks, I can see why my approach was flawed. It was cumulative - in that it tried to capture the possibility of a match on the 2nd (or 3rd, 4th, etc.) but never really adjusted for an earlier match in its "incremental probability". As a result, the numbers are overstated.

I tried your approach in a spreadsheet and agree you both get 23 but the numbers are slightly different. Yours is 49.27 and the exponential method is 49.92. You don't overstate the odds as much as I did but I think it still lacks an adjustment to reflect the match could happen with the 2nd person or the last...

Rounding wouldn't cause that much of a difference. but depending on the computer, this is 364/365 in a calculator and a spreadsheet (32 bit).

0.99726027397260273972602739726027

0.997260273972603

The more I've looked at, the more the exponential "pairs" approach makes sense.

BUt did either of you account for leap year babies?