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2.3.15 Suppose that a basketball player sinks a basket from a certain position on the court with probability $0.35$.

(a) What is the probability that the player sinks three baskets in 10 independent throws?

(b) What is the probability that the player throws 10 times before obtaining the first basket?

(c) What is the probability that the player throws 10 times before obtaining two baskets?

Solution:

(a) $X$ ~ $Binomial(n = 10, \theta = 0.35)$

$$P(X = 3) = {10\choose3}(0.35)^3(0.65)^7$$

(b) $X$ ~ $NegativeBinomial(r = 1, \theta = 0.35)$

$$P(X = 10) = {10\choose10}(0.35)^1(0.65)^{10} \text{[Not right]}$$

(c) $X$ ~ $NegativeBinomial(r = 2, \theta = 0.35)$

$$P(X = 10) = {11\choose10}(0.35)^2(0.65)^{10} \text{[Not right]}$$

Could someone help me understand b and c? Thanks!

Bas
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  • (B) is a geometric random variable. (C) is similar except the tenth attempt must be the second made basket. First use a geometric random variable for B, it’s a bit easier. – Hanzy Jul 05 '18 at 02:37
  • Reference this post for (c): here – Hanzy Jul 05 '18 at 02:41
  • I am not sure the exact intention of questions (b) and (c). I could see how it might be interpreted for (b) as "The player throws ten throws and they are all misses, meaning that the first basket occurs on some throw after the tenth throw" or even "the first ten throws are misses and the eleventh throw is a basket" but I could also see how it might be interpreted as "The first basket occurs on the tenth throw." Each have similar methods for solutions, but will give slightly different answers. – JMoravitz Jul 05 '18 at 03:12

1 Answers1

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b is wrong because you calculated the chance he misses $10$ and then hits number $11$. The question did not require hitting number $11$.

For c you should just compute the chance he gets $0$ or $1$ baskets of the first $11$. In that case he misses $10$ before he makes $2$.

You did not explain your logic for the solutions, but it is clear you are not computing the correct thing.

Ross Millikan
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  • Isn't {b} the same as the first success on the 11th throw? I interpret the question (10 throws BEFORE the first basket) as 10 misses before the first success. To me this isn't at least 10 misses or 10 throws to get the first. – Phil H Jul 05 '18 at 03:34
  • @PhilH: no,I am reading b to ask for at least $10$ throws before success, so starting with $11, 12, \ldots $ misses counts. OP and you are reading it to be exactly $10$ misses, but OP got that right and reports the answer does not check. – Ross Millikan Jul 05 '18 at 03:48
  • One of my goals when writing test items was to make it 100% unambiguous. Even then, students would still misinterpret the question. Math questions that test the finer points of language skills are examples of how not to write a question. – Phil H Jul 05 '18 at 04:09