Math Puzzles (from FiveThirtyEight - new puzzle every Friday)

[Ignoratio Elenchi 2,446]

 

 

 

 

A basketball player is in the gym practicing free throws. He makes his first shot, then misses his second. This player tends to get inside his own head a little bit, so this isnt good news. Specifically, the probability he hits any subsequent shot is equal to the overall percentage of shots that hes made thus far. (His neuroses are very exacting.) His coach, who knows his psychological tendency and saw the first two shots, leaves the gym and doesnt see the next 96 shots. The coach returns, and sees the player make shot No. 99. What is the probability, from the coachs point of view, that he makes shot No. 100?

 

 

From here: Will The Neurotic Basketball Player Make His Next Free Throw?

 

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Edited June 10, 2016 by Ignoratio Elenchi

[John Bender 6,527]

6

[kevzilla 7,964]

If he made the first shot, he should have made them all, right?

Edited January 18, 2016 by kevzilla

[Short Corner 1,741]

A basketball player is in the gym practicing free throws. He makes his first shot, then misses his second. This player tends to get inside his own head a little bit, so this isnt good news. Specifically, the probability he hits any subsequent shot is equal to the overall percentage of shots that hes made thus far. (His neuroses are very exacting.) His coach, who knows his psychological tendency and saw the first two shots, leaves the gym and doesnt see the next 96 shots. The coach returns, and sees the player make shot No. 99. What is the probability, from the coachs point of view, that he makes shot No. 100?

From here: http://fivethirtyeight.com/features/will-the-neurotic-basketball-player-make-his-next-free-throw/

Seems pretty inutitive. Did a 3rd degree probablility tree to check. Figure it will telescope out to 100.

[Buzzbait 1,190]

2/3?

[shadyridr 14,318]

-5^2

[belljr 11,270]

50/50

[Peyton Marino 4,780]

yea ill go 2/3 on based on bayesian something or other

[Sinn Fein 34,146]

The coach has no idea - the player no longer is beholden to the probability of his prior shots.

Player makes the first shot - the probability that he hit the 2nd shot should have been 100% He missed the shot, thus breaking the neurotic spell.

[Sweet J 3,288]

If he made the first shot, he should have made them all, right?

The rule didn't kick in until he was 1 for 2. At that point in the timeline, the parameters of the problem start.

As far as I can tell, there is exactly 50% chance of making the next shot. If he makes that, then he'll have a 66% chance of hitting the one after that.

If he misses it, he'll have 33% chance of hitting the one after that.

But that's as far as I got.

[Peyton Marino 4,780]

yea ill go 2/3 on based on bayesian something or other

actually i'll change my guess to (50/99)=50.51%. coach has to account for the previous shots that he knows occurred, but can only use information that he probably went 50% on those shots.

[LAUNCH 1,505]

If he made the first shot, he should have made them all, right?

I think the word made makes this correct. 2/3 if it was, attempted.

[IC FBGCav 5,779]

A basketball player is in the gym practicing free throws. He makes his first shot, then misses his second. This player tends to get inside his own head a little bit, so this isnt good news. Specifically, the probability he hits any subsequent shot is equal to the overall percentage of shots that hes made thus far. (His neuroses are very exacting.) His coach, who knows his psychological tendency and saw the first two shots, leaves the gym and doesnt see the next 96 shots. The coach returns, and sees the player make shot No. 99. What is the probability, from the coachs point of view, that he makes shot No. 100?

From here: http://fivethirtyeight.com/features/will-the-neurotic-basketball-player-make-his-next-free-throw/

66.66667%

[jon_mx 9,114]

From the coaches perspective, he would have to expect the player to hit half of the next 96 shots. So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots. Since he sees the 99th shot made, he would figure the pecentage of the next shot would be 50/99.

[jon_mx 9,114]

The coach has no idea - the player no longer is beholden to the probability of his prior shots.

Player makes the first shot - the probability that he hit the 2nd shot should have been 100% He missed the shot, thus breaking the neurotic spell.

That needs to be better explained.

[Sweet J 3,288]

The coach has no idea - the player no longer is beholden to the probability of his prior shots.

Player makes the first shot - the probability that he hit the 2nd shot should have been 100% He missed the shot, thus breaking the neurotic spell.

But that's not how the question was set up. The probability of him hitting the third shot is 50% exactly. The probability of hitting the first or the second shot are irrelevant.

[Rove! 2,095]

2/3

[Ignoratio Elenchi 2,446]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

[Short Corner 1,741]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

[jon_mx 9,114]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

Which probably takes you back to the 2/3rds answer.

[Sweet J 3,288]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

I've honestly got no idea where to even start in solving this problem.

[Ignoratio Elenchi 2,446]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

Yes he can.

[Leroy Hoard 14,017]

Flip a coin. Unless of course the coin does not flip, then start over.

[jon_mx 9,114]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

He can. Not knowing about that 99th shot, there are millions of possible ways his shooting could have gone. He could have really gone south and hit a missing streak which would drive his makes to near zero. Or he could have gotton lucky and hit a few in a row and his percentage would have sky-rocketted. Since he made that shot, it becomes more probable that he hit a hot streak and his shooting was above 50%.

[JaxBill 6,957]

50% You either make it or you don't

[Short Corner 1,741]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

He can. Not knowing about that 99th shot, there are millions of possible ways his shooting could have gone. He could have really gone south and hit a missing streak which would drive his makes to near zero. Or he could have gotton lucky and hit a few in a row and his percentage would have sky-rocketted. Since he made that shot, it becomes more probable that he hit a hot streak and his shooting was above 50%.

The expectation after the 98th shot is the same regardless.

[Peyton Marino 4,780]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

Yes he can.

at the time when the coach left the gym after the 2nd shot, the player was 50% and thus the coach can only assume he hit 50% of shots 3-98. shot 99 can't affect shots 3-98.

[Ignoratio Elenchi 2,446]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

Yes he can.

at the time when the coach left the gym after the 2nd shot, the player was 50% and thus the coach can only assume he hit 50% of shots 3-98. shot 99 can't affect shots 3-98.

Seeing shot 99 affects the coach's estimation of what happened on shots 3-98.

[SoCalBroncoFan 572]

I am going to go out on a limb here and say it is 0%.

Here's why:

The key phrase in the question is, "from the coach's view". The coach saw the first two shots, they were 50%. He didn't see shots 3-98 so they are irrelevant. Then he saw shot 99, which the player made. Based on the previous two shots that he saw, shots 1 and 2, he should expect the player to miss the shot, which be a 0% probability.

Probably wrong, but a different way of seeing it.

[Dragons 213]

If he made shot 99, then 100% on 100, except he shouldn't have missed #2

Edited January 18, 2016 by Dragons

[Pigskin Fanatic 619]

can't be 50-50 from the op

Specifically, the probability he hits any subsequent shot is equal to the overall percentage of shots that hes made thus far.

there have been 99 shots attempted, first and last were made, 50/99 is a good assumption. the third shot probability was 50-50 and that was the only shot that had a 50-50 chance by this premise. From there forward every other shot will be either one above 50% or one below. i'd go with 50/99 and slightly more likely he makes it than miss.

i like the bronco answer above as well from a different perspective.

[The General 24,969]

If him making it ruins my bet...100%.

[Short Corner 1,741]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

Yes he can.

at the time when the coach left the gym after the 2nd shot, the player was 50% and thus the coach can only assume he hit 50% of shots 3-98. shot 99 can't affect shots 3-98.

Seeing shot 99 affects the coach's estimation of what happened on shots 3-98.

If he understands the psychological tendency his estimation on shots 3-98 should be unaffected by shot 99.

[jon_mx 9,114]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

He can. Not knowing about that 99th shot, there are millions of possible ways his shooting could have gone. He could have really gone south and hit a missing streak which would drive his makes to near zero. Or he could have gotton lucky and hit a few in a row and his percentage would have sky-rocketted. Since he made that shot, it becomes more probable that he hit a hot streak and his shooting was above 50%.

The expectation after the 98th shot is the same regardless.

Let's say you see him make the next 15 shots in a row, would that impact what you think happened during shots 3 thru 98?

[Ignoratio Elenchi 2,446]

If he understands the psychological tendency his estimation on shots 3-98 should be unaffected by shot 99.

I have a bag with two marbles in it, one white and one black. Without looking I randomly pull one out and hide it behind my back. What's the probability that it was the white one? 50%, right?

Now I reach in the bag, pull out the other marble and look at it. It's white. Do you still think there's a 50% probability that the marble behind my back is white? Or has your estimation of the prior event changed now that you've gained additional information?

Before seeing the 99th shot, the coach can't do any better but to assume that the player has shot 50% on his first 98. But after seeing the 99th shot go in, he's gained additional information about those prior shots, because the probability that the 99th shot would be made depends on what happened in the first 98.

[Cjw_55106 5,087]

The coach doesn't give a ####, that's why he left the gym.

[Sweet J 3,288]

I am going to go out on a limb here and say it is 0%.

Here's why:

The key phrase in the question is, "from the coach's view". The coach saw the first two shots, they were 50%. He didn't see shots 3-98 so they are irrelevant. Then he saw shot 99, which the player made. Based on the previous two shots that he saw, shots 1 and 2, he should expect the player to miss the shot, which be a 0% probability.

Probably wrong, but a different way of seeing it.

This post has so much wrong it in I can only conclude that you are a girl.

I'm kidding. Kind of.

[Sweet J 3,288]

If he made shot 99, then 100% on 100, except he shouldn't have missed #2

Look, the percentages can not be either 0% or 100%. Can we please just get that out of the way?

[jhib 2,006]

I'm going with 2/3 he makes it after seeing the 99th shot went in (1/3 if he saw the 99th missed, and 1/2 if he didn't see the 99th shot).

From the coach's perspective, the 3rd shot has a 1/2 chance of going in after seeing the first two shots. If he sees the 3rd shot, then the 4th shot has a 2/3 chance of being the same as the 3rd. But if the coach doesn't see the 3rd shot, he'd have to go with 1/2 chance of making the 4th (since the 2/3 chance he'd make it had he made the 3rd is cancelled out by the 2/3 chance he'd miss it had he missed the 3rd, which was equally as likely). I'm pretty sure it would work the same way whether the number of shots the coach didn't see was 1 or 96.

[Peyton Marino 4,780]

I am going to go out on a limb here and say it is 0%.

Here's why:

The key phrase in the question is, "from the coach's view". The coach saw the first two shots, they were 50%. He didn't see shots 3-98 so they are irrelevant. Then he saw shot 99, which the player made. Based on the previous two shots that he saw, shots 1 and 2, he should expect the player to miss the shot, which be a 0% probability.

Probably wrong, but a different way of seeing it.

what in the world

[jhib 2,006]

50% You either make it or you don't

I'm pretty sure there's 100% chance you either make it or you don't.

[Dave Baker 11]

2/3

[Dickies 9,482]

I am going to go out on a limb here and say it is 0%.

Here's why:

The key phrase in the question is, "from the coach's view". The coach saw the first two shots, they were 50%. He didn't see shots 3-98 so they are irrelevant. Then he saw shot 99, which the player made. Based on the previous two shots that he saw, shots 1 and 2, he should expect the player to miss the shot, which be a 0% probability.

Probably wrong, but a different way of seeing it.

That's one way to put it.

[Sweet J 3,288]

I am going to go out on a limb here and say it is 0%.

Here'

s why:

The key phrase in the question is, "from the coach's view". The coach saw the first two shots, they were 50%. He didn't see shots 3-98 so they are irrelevant. Then he saw shot 99, which the player made. Based on the previous two shots that he saw, shots 1 and 2, he should expect the player to miss the shot, which be a 0% probability.

Probably wrong, but a different way of seeing it.

what in the world

[jon_mx 9,114]

I am going to go out on a limb here and say it is 0%.

Here's why:

The key phrase in the question is, "from the coach's view". The coach saw the first two shots, they were 50%. He didn't see shots 3-98 so they are irrelevant. Then he saw shot 99, which the player made. Based on the previous two shots that he saw, shots 1 and 2, he should expect the player to miss the shot, which be a 0% probability.

Probably wrong, but a different way of seeing it.

what in the world

He actually nailed it. Well at least the part where he said he was probably wrong.

[jon_mx 9,114]

So at that point the best assumption the coach could have made is that the player has made 49 of his 98 shots.

He can make a better assumption than that, since he's seen the player make his 99th shot.

No he can't

Yes he can.

at the time when the coach left the gym after the 2nd shot, the player was 50% and thus the coach can only assume he hit 50% of shots 3-98. shot 99 can't affect shots 3-98.

No it doesn't affect shots 3-98. But it does tell you something abou them. If he would have missed it, it would have told you something different.

Edited January 18, 2016 by jon_mx

[tommyboy 4,621]

Zero

[IC FBGCav 5,779]

Something tells me it is over 66.67% but can't put my finger on it.

[IC FBGCav 5,779]

I am not going to run the math but what we do know is that the 3rd shot had a 50/50 shot of going in. If the 3rd shot went in the probabilities go to higher percentages approaching shot 99. If he misses the 3rd shot the same it true in reverse. So does this information shed any light to why shot 99 went in? If so, it probably means that he made shot 3 and shot 100 has a very good chance of going in.

Edited January 18, 2016 by FatUncleJerryBuss

[cap'n grunge 7,666]

It depends. Did Belichick have access to the balls?