Question

Sophie is a dog that loves to play catch. Unfortunately, she isn't very good, and the probability that she catches a ball is only .1. Let \(x\) be the number of tosses required until Sophie catches a ball. a. Does \(x\) have a binomial or a geometric distribution? b. What is the probability that it will take exactly two tosses for Sophie to catch a ball? c. What is the probability that more than three tosses will be required?

Short answer

a. \(x\) has a geometric distribution. b. The probability that it will take exactly two tosses for Sophie to catch a ball is 0.09. c. The probability that more than three tosses will be required is 0.729.

Step-by-step solution

Identifying the Distribution

The prompt states that we're looking for the number of tosses required until Sophie catches a ball; we're not concerned with the number of successes within a given set of trials. This scenario describes a geometric distribution and not a binomial distribution. Therefore, \(x\) has a geometric distribution.

Calculating the Probability for Exact Two Tosses

The probability that Sophie catches the ball on her second toss is the product of the probability that she misses the ball on her first toss and the probability that she catches it on the second toss. Since these are independent events, we can use the formula for the geometric probability mass function, \(P(X = x) = (1 - p)^{(x - 1)} * p\), which gives \(P(X = 2) = (1 - 0.1)^{(2 - 1)} * 0.1 = 0.09\).

Calculating the Probability for More Than Three Tosses

The problem's asking for the probability that more than three tosses will be required. In other words, at least four tosses. We'll find the probability of the first three tosses, and then subtract that from 1 (representing the total probability). Thus, \(P(X > 3) = 1 - [P(X = 1) + P(X = 2) + P(X = 3)]\). By putting the probabilities for X=1, 2 and 3 in the formula, \(P(X > 3) = 1 - [(0.1) + (0.9*0.1) + (0.9^2*0.1)] = 0.729\).