[SOLVED] Binomial Distribution Problem

**akolman** · October 23rd 2009, 06:08 PM

Hello, I am lost with the following problem.

Consider two players, player A and player B, in a game that involves tossing a (not necessarily fair) coin with $p = P(H)$ such that A gains one dollar when a head $(H)$ occurs and B gains one dollar if a $T (tail)$ comes up. Assume that player A needs $n$ while player B needs $m$ more dollars to win the game (both $n$ and $m$ are non-negative integers)

Let $A$ be the event that $n$ heads occur before $m$ tails. $A$ occurs if and only if there are at least $n$ heads in the fi…rst $n +m- 1$ trials. Explain why this reasoning is correct. Then use the binomial distribution to write an expression for $P(A) = P_{n,m}.$

I know it is a very long and wordy problem, any hint will be appreciated, Thanks.

**novice** · October 24th 2009, 06:02 AM

Originally Posted by akolman

Hello, I am lost with the following problem.

Consider two players, player A and player B, in a game that involves tossing a (not necessarily fair) coin with $p = P(H)$ such that A gains one dollar when a head $(H)$ occurs and B gains one dollar if a $T (tail)$ comes up. Assume that player A needs $n$ while player B needs $m$ more dollars to win the game (both $n$ and $m$ are non-negative integers)

Let $A$ be the event that $n$ heads occur before $m$ tails. $A$ occurs if and only if there are at least $n$ heads in the fi…rst $n +m- 1$ trials. Explain why this reasoning is correct. Then use the binomial distribution to write an expression for $P(A) = P_{n,m}.$

I know it is a very long and wordy problem, any hint will be appreciated, Thanks.

The question is not correct because there are contradictions between the statements 1 and 2 ; and statements 2 and 3, as labeled below:

Statement 1: Assume that player A needs $n$ while player B needs $m$ "more" dollars to win the game (both $n$ and $m$ are non-negative integers)

Statement 2: $A$ occurs if and only if there are at least $n$ heads in the fi…rst $n +m- 1$ trials.

Statement 3: Then use the binomial distribution to write an expression for $P(A) = P_{n,m}.$

In Statement 1, the word "more" should have been struck out.
P(A) in Statement 3 should have been $P(A)=\binom{n+m-1}{n}p_h^np_t^{m-1}$

-------------------
Assuming A needs to win $n$ dollars, and B, $m$ dollars, A will occur, if and only if there are $n$ heads in the first $n+m-1$ trials.

Example: Say n=3 dollars, m=2 dollars. For A to win there are n+m-1 trials, which have n+m-1=4 possible sample spaces {HHH},{HHTH},{HTHH},{THHH}

$P(A)=\binom{3+2-1}{3}p_h^3p_t^{3+2-1-3}=\binom{4}{3}p_h^3p_t^{1}$

Note: This is not a fair question because it does not tell us who will go first in the game. There is a big difference as to who will go first because the first to go will have a much better probability of winning. This you can test yourself by considering the expected value of each player. A flaw is found in this question in which it expected the solution to align with binomial distribution, for for which after a string of $n$ number of heads appeared, B would bother to complete the remaining number trials.

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