Chapter 4: Q. 4.31 (page 172)

A jar contains $m + n$ chips, numbered $1, 2, \dots, n + m$ . A set of size $n$ is drawn. If we let $X$ denote the number of chips drawn having numbers that exceed each of the numbers of those remaining, compute the probability mass function of $X$ .

Short Answer

Expert verified

$P (X = k) = \frac{(\begin{matrix} n + m - k - 1 \\ n - k \end{matrix})}{(\begin{matrix} n + m \\ n \end{matrix})}$

Step by step solution

Given information

A jar contains $m + n$ chips, numbered $1, 2, \dots, n + m$ . A set of size $n$ is drawn.

Explanation

Observe that $X \in {0, \dots, n}$ . Take any $k \in {0, \dots, n}$ . Let's calculate $P (X = k)$ . Observe that there are $(\begin{matrix} n + m \\ n \end{matrix})$ of all possible combinations of taken chips. If $X = k$ , that means that we have taken $k$ largest number, have not taken $(k + 1) s t$ largest number and all other remaining $n - k$ numbers out of remaining $n + m - k - 1$ numbers have been taken freely.

Final answer

The probability mass function is

$P (X = k) = \frac{(\begin{matrix} n + m - k - 1 \\ n - k \end{matrix})}{(\begin{matrix} n + m \\ n \end{matrix})}$

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A jar contains $m + n$ chips, numbered $1, 2, \dots, n + m$ . A set of size $n$ is drawn. If we let $X$ denote the number of chips drawn having numbers that exceed each of the numbers of those remaining, compute the probability mass function of $X$ .

Short Answer

Step by step solution

Given information

Explanation

Final answer

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A jar contains m+nchips, numbered 1,2,…,n+m. A set of size nis drawn. If we let X denote the number of chips drawn having numbers that exceed each of the numbers of those remaining, compute the probability mass function of X.

Short Answer

Step by step solution

Given information

Explanation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Mechanics Maths

Applied Mathematics

Decision Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

A jar contains $m + n$ chips, numbered $1, 2, \dots, n + m$ . A set of size $n$ is drawn. If we let $X$ denote the number of chips drawn having numbers that exceed each of the numbers of those remaining, compute the probability mass function of $X$ .