An urn contains n white and m black balls, where n and m

Chapter 2, Problem 29P

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QUESTION:

Problem 29P

An urn contains n white and m black balls, where n and m are positive numbers.

(a) If two balls are randomly withdrawn, what is the probability that they are the same color?

(b) If a ball is randomly withdrawn and then replaced before the second one is drawn, what is the probability that the withdrawn balls are the same color?

(c) Show that the probability in part (b) is always larger than the one in part (a).

Questions & Answers

QUESTION:

Problem 29P

An urn contains n white and m black balls, where n and m are positive numbers.

(a) If two balls are randomly withdrawn, what is the probability that they are the same color?

(b) If a ball is randomly withdrawn and then replaced before the second one is drawn, what is the probability that the withdrawn balls are the same color?

(c) Show that the probability in part (b) is always larger than the one in part (a).

ANSWER:

Solution :

Step 1 of 3:

Given ‘n’ white and ‘m’ black balls.

Here n and m are positive numbers.

Our goal is:

a). We need to find the probability that they are the same color.

b). We need to find the probability that the withdrawn balls are the same color.

c). We need to show that  the probability in part (b) is always larger than the one in part (a).

a). Given the 2 balls randomly withdrawn, then the probability of 2 white ball is

Then the probability of 2 black balls is

 

Let the probability of 2 balls of the same color is

         =

Therefore, the probability of 2 balls of the same color is .


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