My friend and I make a bet. I get 1 minute to shoot a basketball from 1 specific spot beyond the three point line. If I make 5, I get $500. I get an additional $150 for every shot I make after 5. If I don't make 5, I have to pay $200. Just so I'm not leaving any variables out, let's say I'm a 25% three point shooter and I'm gonna get a shot off approximately every 3 seconds.

What is the $ value of the first shot I make? How about the first shot I take?

Thanks in advance for your responses!

According to your assumptions you get 20 shots in, with 25% chance each. This kind of scenario (independent events with repeated trials) is governed by a binomial distribution. Putting your numbers into a binomial calculator, you get:

58.5% chance of success (5 or more hits).

You can calculate the dollar value easily from that. So the bet is very much in your favor if your assumptions are correct.

If your odds of making a shot are only 15% instead of 25% (nerves?), the odds of you winning is only 17%. If you only make one shot every four seconds instead of every three, the odds of you winning drop to 31%. If you only take one shot every four seconds AND only hit 15%, the odds go down to 6%.

So the outcome is highly dependent on your ability to perform consistently - small deviations from your assumptions and it becomes a losing bet very quickly.

It depends on when you make it. If you are on your last shot and you still haven't made it, your shot is worth -200. But if you make your first shot, it's worth more.

This seems to be asking for the EV of playing. You get 20 shots and you shoot 25%. It looks like just a straightforward calculation:

EV = -200 * (20 choose 0) * (1/4)^0 * (3/4)^20 - 200 * (20 choose 1) * (1/4)^1 * (3/4)^19 + ... + 500 * (20 choose 5) (1/4)^5 * (3/4)^15 + 650 * (20 choose 6) * (1/4)^6 * (3/4)^14 + ...

I'd guess this is +EV without doing the calculation because of the high payoff on the back end.

Assuming your 25% success rate is constant and you shoot 60/3 = 20 times in the one minute trial, the binomial distribution applies.

For example, the chance you make exactly 3 baskets is 6.7%. If you do, you lose $200. The chance of 6 baskets is 16.9% and you will gain $650.

Doing this for the chance of 0, 1, 2, ..., 19,20 baskets and factoring in the win and loss for each, your expected gain is $324.42.

Not too bad of a deal, I would say. Of course, if you miss the first four shots, you may feel the pressure and the 25% success rate may drop. On the other hand, if you make 2 or 3 of the first four, the basket diameter may seem like a kids wading pool and your success rate will increase. So, it may just balance out.

Can you get the probabilities for getting exactly 5, 6, 7 and 8 hits? That should probably cover good enough for how much of the $150s he's getting.

P(X=5): 20.2%
P(X=6): 16.9%
P(X=7): 11.2%
P(X=8): 6.1%
P(X=9): 2.7%
P(X=10 or more): 1.4%

OP can do the math from there.

Your first question is simply the difference in EV between 19 and 20 shots. Your second question is fuzzy. One way to interpret it would be the difference in EV of your actual results and the EV if you called on of your hits a miss.

Are you suggesting there is such a thing as a "hot hand?"

FYP

For the 20 throws: -0.415x$200 + .202x$500 + .169x$650 + .112x$800 + .061x$950 + .027x$1100 + 0.014x$1250 + (skip the rest)~= $(-83+101+110+90+58+30+18+skip the rest)~=$324. So every throw is on average worth approx $324/20 $16.2, let's say $17 because of the part we skipped.

Are you suggesting that someones shooting % is permanently fixed?

(links work only if you copy all the text and then paste on browser not by clicking)

http://www.wolframalpha.com/input/?i=NSum[20!%2F%28n!*%2820-n%29!%29*%281%2F4%29^n*%283%2F4%29^%2820-n%29%2C{n%2C0%2C4}]




0.41484 chance to lose.

http://www.wolframalpha.com/input/?i=NSum[%28500%2B%28n-5%29*150%2920!%2F%28n!*%2820-n%29!%29*%281%2F4%29^n*%283%2F4%29^%2820-n%29%2C{n%2C5%2C20}]-NSum[200*20!%2Fn!%2F%2820-n%29!*%281%2F4%29^n*%283%2F4%29^%2820-n%29%2C{n%2C0%2C4}]

EV +323.4


The game stops being profitable if your true probability of 3pointer success is smaller than ~17.3%.


If you miss the first shot the EV goes from +323.4 to +268.0 a drop of 55.4.
If you make it though the EV goes from 323.4 to 489.8 a rise of 166.4.

Now if you ask the first shot you make successfully as noted before it depends on when that happens. If you lose the first 4 times and make the 5th it will be different than if you lose the first 2 and make the 3rd and if you make the first with only 3 shots left its pointless anyway. So it starts from very important +166.4 (first one) to eventually worthless.

Guys,

First, thanks for your answers. Here is what I gather from piecing together the information you gave me. it seems that the first shot I make is dependent upon which shot I make it on, which would alter the initial variables I gave before entering them into a binomial distribution calculator.

The first shot I take should be the EV of the entire bet/20

Is this correct?

Is this homework or something? Who cares about "make" or "take" unless you're doing homework?

Didace, with all due respect, the answer to this is very similar to one of the, if not the most important concept in poker. There is no such thing as a "hot hand" per se, but there is such thing as streaks of randomization and there is such thing as averages.

Mr. Slansky,

This is what I gathered from another thread I started elsewhere in the site: It seems that the first shot I make is dependent upon which shot I make it on, which would alter the initial variables I gave before entering them into a binomial distribution calculator.

The first shot I take should be the EV of the entire bet/20

However, I am very interested in hearing an explanation of how your response answers my question.

People like to think about stuff sometimes.

Probability forum, and don't double post.

I agree with the $323.42 for 20 shots. But with a shot every 3 seconds you could get 21 shots if the last one at the buzzer counts. That brings it to $378.56. With 20 shots your break even percentage is about 17.12%, and with 21 shots your break even percentage is about 16.27%.

Here's an R function where you can play with the number of shots and the win percentage. R is a free and easy download.

Call like this:

ft(.25,20)
[1] 323.422221148394
> ft(.25,21)
[1] 378.560619405039
> ft(.1712,20)
[1] 0.24005271663464
> ft(.1711,20)
[1] -0.10310417143894
> ft(.1628,21)
[1] 0.207547065750902
> ft(.1627,21)
[1] -0.151889870266697

I think Didace has a firm grasp of that concept, but cards are a bad analogy for basketball shots. Shooting is a skill (whereas catching a card is not), and a human's performance can vary. There absolutely can be a such thing as a non-random "hot hand" or even a cold hand, because there can be causes for it (mental and perhaps others).

This isn't to say that the assumption of a perfectly binomial distribution isn't a good approximation.

Haha re buzzer. But the problem of course is that technically if a shot takes 3sec you get the ball at exactly 60.00 for the 21st trial and it takes a time to form a jump shot or just shot that is nonzero so the buzz hits before ball leaves your hands in a situation of precisely 3 sec each. Also the chance to make it is a function of how much cushion you have to release from hands anyway (desperate vs confident).


Given that all this is technically bs (i mean its never exactly 3 sec etc) we should in fact model this differently with a probability distribution of time taken per shot. Then i suppose it becomes a much harder to do "exactly" problem even if we can do it as a Normal 3sec +- 0.3 sec each tirial. But that would be the proper way to account for last sec chances etc due to prior time taken fluctuations where the last shot doesnt need 3 sec room, all it takes is a 0.3 sec cushion to form the shot and release and time expires only if the ball is still in your hands before it leaves.

Even more realistic should be to model how a human (or that particular human) responds to stress or success influencing make the shot probability. Stress may also influence time per shot duration. I wonder what is the most efficient way to deal with stress then. It would depend on what happens to say 25% when you try to do in 2.5 vs 3 or 3.3sec. Its in fact interesting anyway but only if it can be modeled with solid data how it all becomes fluid and a function of the overall progress/history.

I thought that the shot counts if the ball was in forward motion at the buzzer, even if it was still in the player's hand, but according to wiki it has to be in mid-air:

-wiki

Would have to consult official rules to be sure, and it could be different in NCAA vs. NBA, etc.

Even if it worked that way, there would still be reaction time unless he starts it in motion before the start whistle.


Studies have shown that there is no such thing as a "hot hand", and probabilities of making shots don't depend on making previous shots. That's in real games though, not 1 minute 3-point drills. I would think you can settle into a grove and have a much higher probability.

As a matter of principle it should matter if the ball is in the hands and a release must be made before the expiration to be proper. The idea is that at least in principle by still being in contact with the ball you influence the result so how can it be tolerated rationally to have the final influence by the person shooting take place after the expiration? So without knowing the exact rules i would want them to be that as long as the shooter releases before buzzer or at the buzzer within observation error (+-0.05sec?) its good.

By the way 0.05s is the time it usually takes to touch with your index (your nail) a surface twice if you move very close to human tolerance for such effort. So imagine how much more time it takes to actually grab a ball and try to form any kind of trajectory, at least 0.15s it seems under the best of conditions and probably more like 0.3 sec or more.

Then you should publish these findings. It would be pretty ground-breaking stuff if you could demonstrate something like a hot hand.

Permanently fixed is a pretty strong term. It is not the opposite of a "hot hand" if that is what you are trying to imply.

But are you ever really "not in contact" with the ball?

Maybe I was thinking about football where the quarterback's arm has to be coming forward to count as a pass.

I agree.

Good points and interesting to think about. I've typed a few different answers in response and I think there's a solid counterargument, but I'm a few bloody Mary's down and the airport and I'm not sure I can formulate it at the moment.