Q.4.23 Balls are randomly withdrawn, on... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

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Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N white and M black balls.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, $n 鈮� N, m 鈮� M$

Short Answer

Expert verified

In the given information the required probability is $P (X 鈮� n) =$ ${鈭�}_{k = n}^{n + m - 1} \frac{(\begin{array}{l} N \\ k \end{array}) (\begin{matrix} M \\ n + m - 1 - k \end{matrix})}{(\begin{matrix} N + M \\ n + m - 1 \end{matrix})}$

Step by step solution

Given information

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

Step 2:Calculation

$P (X 鈮� n) = {鈭�}_{k = n}^{n + m - 1} P (X = n)$

= ${鈭�}_{k = n}^{n + m - 1} \frac{(\begin{array}{l} N \\ k \end{array}) (\begin{matrix} M \\ n + m - 1 - k \end{matrix})}{(\begin{matrix} N + M \\ n + m - 1 \end{matrix})}$

Final answer

The required probability is $P (X 鈮� n) =$ ${鈭�}_{k = n}^{n + m - 1} \frac{(\begin{matrix} N \\ k \end{matrix}) (\begin{matrix} M \\ n + m - 1 - k \end{matrix})}{(\begin{matrix} N + M \\ n + m - 1 \end{matrix})}$

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91影视

Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N white and M black balls.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, $n 鈮� N, m 鈮� M$

Short Answer

Step by step solution

Given information

Step 2:Calculation

Final answer

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91影视

Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N white and M black balls.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鈮�N,m鈮�M

Short Answer

Step by step solution

Given information

Step 2:Calculation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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, $n 鈮� N, m 鈮� M$