/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 Let \(X\) denote the number of w... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 71

Let \(X\) denote the number of white balls selected when \(k\) balls are chosen at random from an urn containing \(n\) white and \(m\) black balls.

Short Answer

Expert verified
To find the probability of selecting exactly X white balls when choosing k balls at random from an urn containing n white balls and m black balls, use the formula: \[P(X) = \frac{\binom{n}{X} * \binom{m}{k - X}}{\binom{n + m}{k}}\]

Step by step solution

01

Find the total number of ways to choose k balls from the urn.

First, let's find the total number of ways to choose k balls from a set of n white balls and m black balls. This can be done using the combination formula: \[\binom{n + m}{k} = \frac{(n + m)!}{k!(n+ m - k)!}\] This represents the total possible ways of choosing k balls out of (n+m) balls, without considering the color of the balls.
02

Find the number of ways to select X white balls.

Now, we have to find the number of ways to select exactly X white balls out of n white balls. We can do this using the combination formula: \[\binom{n}{X} = \frac{n!}{X!(n - X)!}\] This represents the number of ways to choose X white balls out of n white balls.
03

Find the number of ways to select (k-X) black balls.

Similarly, since we have picked X white balls, we should pick (k-X) black balls out of m black balls. We can find the number of ways to do this with the combination formula: \[\binom{m}{k - X} = \frac{m!}{(k - X)!(m - k + X)!}\] This represents the number of ways to choose (k - X) black balls out of m black balls.
04

Find the probability of selecting X white balls and (k-X) black balls.

Now that we have found the number of ways to select X white balls and (k-X) black balls, we can calculate the probability of this specific selection. To calculate this probability, divide the product of the number of ways to select X white balls and (k-X) black balls by the total number of ways to select k balls from the urn: \[P(X) = \frac{\binom{n}{X} * \binom{m}{k - X}}{\binom{n + m}{k}}\] This formula gives us the probability of selecting exactly X white balls when choosing k balls at random from an urn containing n white balls and m black balls.

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Most popular questions from this chapter

Suppose three fair dice are rolled. What is the probability that at most one six appears?

Show that $$ \lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2} $$ Hint: Let \(X_{n}\) be Poisson with mean \(n\). Use the central limit theorem to show that \(P\left[X_{n} \leq n\right] \rightarrow \frac{1}{2}\)

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