Question

researchers recorded data on pet ownership for randomly selected households in a large city. the venn diagram summarizes the data using events c: has a cat and d: has a dog.
if a household that owns a cat is selected at random, what is the probability that the household owns a dog?
(round to 4 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Find the probability of owning a cat

The probability of owning a cat \(P(C)\) is the sum of the probability of only owning a cat and the probability of owning both a cat and a dog. Let \(x\) be the probability of owning both. The total probability of all events is \(1\). So \(P(C)=x + 0.16\). Also, \(0.22+x + 0.16+0.46=1\), which gives \(x=1-(0.22 + 0.16+0.46)=0.16\). Then \(P(C)=0.16 + 0.16=0.32\).

Step2: Use the formula for conditional probability

The formula for conditional probability is \(P(D|C)=\frac{P(D\cap C)}{P(C)}\). We know \(P(D\cap C) = 0.16\) (from Step 1) and \(P(C)=0.32\). So \(P(D|C)=\frac{0.16}{0.32}=0.5\)

Answer:

\(0.5000\)