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a. An urn contains n white and m black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability n/(n + m), they are all white. Hint: Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn. b. A pond contains 3 distinct species of fish, which we will call the Red, Blue, and Green fish. There are r Red, b Blue, and g Green fish. Suppose that the fish are removed from the pond in a random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Red fish are the first species to become extinct in the pond? Hint: Write P{R} = P{RBG} + P{RGB), and compute the probabilities on the right by first conditioning on the last species to be removed.

a. An urn contains n white and m black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability n/(n + m), they are all white. Hint: Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn. b. A pond contains 3 distinct species of fish, which we will call the Red, Blue, and Green fish. There are r Red, b Blue, and g Green fish. Suppose that the fish are removed from the pond in a random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Red fish are the first species to become extinct in the pond? Hint: Write P{R} = P{RBG} + P{RGB), and compute the probabilities on the right by first conditioning on the last species to be removed.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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How can I solve that question 3.7 (a and b)?

## Problem 3.7 ### a. Probability with Balls in an Urn An urn contains \( n \) white and \( m \) black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability \( \frac{n}{n + m} \), they are all white. **Hint:** Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn. ### b. Probability of Fish Extinction A pond contains 3 distinct species of fish, which we will call the Red, Blue, and Green fish. There are \( r \) Red, \( b \) Blue, and \( g \) Green fish. Suppose that the fish are removed from the pond in a random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Red fish are the first species to become extinct in the pond? **Hint:** Write \( P\{R\} = P\{RBG\} + P\{RGB\} \), and compute the probabilities on the right by first conditioning on the last species to be removed.
Transcribed Image Text:## Problem 3.7 ### a. Probability with Balls in an Urn An urn contains \( n \) white and \( m \) black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability \( \frac{n}{n + m} \), they are all white. **Hint:** Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn. ### b. Probability of Fish Extinction A pond contains 3 distinct species of fish, which we will call the Red, Blue, and Green fish. There are \( r \) Red, \( b \) Blue, and \( g \) Green fish. Suppose that the fish are removed from the pond in a random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Red fish are the first species to become extinct in the pond? **Hint:** Write \( P\{R\} = P\{RBG\} + P\{RGB\} \), and compute the probabilities on the right by first conditioning on the last species to be removed.
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