Questions: Suppose that 24% of people have dogs. If two people are randomly chosen, what is the probability that they both have a dog? Write your answer as a percent and round to the nearest hundredth of a percent. There is a % chance the two randomly chosen people both have dogs.

Solution Steps

We are tasked with finding the probability that both of two randomly chosen people have dogs, given that \( 24\% \) of people have dogs. This can be modeled using a binomial distribution where:

• \( n = 2 \) (the number of trials, or people chosen),
• \( x = 2 \) (the number of successes, or people with dogs),
• \( p = 0.24 \) (the probability of success, or having a dog),
• \( q = 1 - p = 0.76 \) (the probability of failure, or not having a dog).

The probability of exactly \( x \) successes in \( n \) trials is given by the formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Substituting the values, we have:

\[ P(X = 2) = \binom{2}{2} \cdot (0.24)^2 \cdot (0.76)^{2-2} \]

Calculating the components:

• \( \binom{2}{2} = 1 \)
• \( (0.24)^2 = 0.0576 \)
• \( (0.76)^0 = 1 \)

Thus,

\[ P(X = 2) = 1 \cdot 0.0576 \cdot 1 = 0.0576 \]

To express the probability as a percentage, we multiply by \( 100 \):

\[ P(X = 2) \times 100 = 0.0576 \times 100 = 5.76\% \]

Final Answer