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Let's say there is a certain basketball player who has an accuracy of 50% for shooting his shots. Let's imagine he plays games where his average shots taken is 20. What's the probability he hits AT LEAST 5 out of 20 of those shots in a row?

I know that hitting 5 shots in a row is simply 0.5^5, but the difference here is that the player is shooting out of a certain number of shots total. How would you solve this? Would it be relatively easy math?

I'm taking a Statistics class and a problem similar to this popped up. We are trying to do simulations of random numbers to estimate the percentage, but I wanted to find out the exact value. Any tips to get started? Please explain all reasoning :) Thanks!

Freedom
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1 Answers1

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I take it that you mean at least one streak of $5$ or more somewhere in the $20$ shots.

Let $P_n$ denote the probability of getting $5$ successive hits (streak) by the $n_{th}$ trial

then $P_{n} - P_{n-1}$ means that you get a streak for the first time on the $n_{th}$ trial only,

which means that you had not got a streak on the $(n-6)_{th}$ trial, and also missed the shot after that, e.g. for first getting the streak on the $8_{th}$ trial, you didn't have a streak on the $2_{nd}$ , missed the shot on the $3_{rd}$ trial, and then got a streak of $5$.

Using the usual notation of $p$ and $q$ for probabilities of success/failure for individual trials, $P_{n} - P_{n-1} = [1 - P_{n-6}]qp^5$ for $n\ge6$

Starting conditions are: $P_0\;\; thru\;\; P_4= 0, P_5 = (1/2)^5, P_6 = P_5+[1- P_0] *(1/2)^6, ...$

Proceed.....

true blue anil
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