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Chapter 4: Q.4.23 (page 174)

Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N white and M black balls. Find the probability that n white balls are drawn before m black balls,n≤N,m≤M

Short Answer

Expert verified

In the given information the required probability isP(X≥n)=∑k=nn+m-1NkMn+m-1-kN+Mn+m-1

Step by step solution

01

Given information

Consider this idea. There will be drawn nwhite balls before mblack balls if and only if there are at least nwhite balls within n+m-1drawn balls. This is because the fact that in that case, there is mor less black balls within n+m-1drawn balls, so we have satisfied our condition. If we mark with X the number of white balls drawn withinn+m-1drawn balls, we have that X has Hypergeometric distribution.

02

Step 2:Calculation

P(X≥n)=∑k=nn+m-1P(X=n)

= ∑k=nn+m-1NkMn+m-1-kN+Mn+m-1

03

Final answer

The required probability isPX≥n=∑k=nn+m-1NkMn+m-1-kN+Mn+m-1

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Most popular questions from this chapter

Let X be a binomial random variable with parameters (n, p). What value of p maximizes P{X = k}, k = 0, 1, ... , n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by choosing that value of p that maximizes P{X = k}. This is known as the method of maximum likelihood estimation.

A student is getting ready to take an important oral examination and is concerned about the possibility of having an “on” day or an “off” day. He figures that if he has an on the day, then each of his examiners will pass him, independently of one another, with probability8, whereas if he has an off day, this probability will be reduced to4. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on the day, should he request an examination with3examiners or with5examiners?

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(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4possibilities is equally likely, what are the possible values of Xand what are their associated probabilities?

(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1or 2 fingers, what are the possible values ofX and their associated probabilities?

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(a) Find PEi.

(b) Give an approximation for the probability that no3individuals share the same birthday.

(c) Evaluate the preceding when n=88(which can be shown to be the smallest value ofnfor which the probability exceeds.5).

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