Use the table to answer the question.

Bowling Dancing Movies Total
20–25 Years Old 15 13 12 40
26–30 Years Old 12 10 8 30
31–35 Years Old 18 11 21 50
Total 45 34 41 120
In a survey, 120 people are asked how they like to spend their free time. Their responses are recorded on the table based on their age. What is the probability that a person spends their time going to the movies, given that they are 31–35 years old? Express your answer as a percentage, rounded to the nearest whole number.

1 answer

To find the probability that a person spends their time going to the movies given that they are 31–35 years old, we can use the following formula for conditional probability:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

where:

  • \(P(A|B)\) is the probability of event A occurring given event B is true.
  • \(P(A \cap B)\) is the probability of both events A and B occurring.
  • \(P(B)\) is the probability of event B occurring.

In this scenario:

  • Event A: Spending time going to the movies.
  • Event B: Being 31–35 years old.

From the table, we can see the data for people aged 31–35:

  • Total people aged 31–35: 50
  • People aged 31–35 who like going to the movies: 21

Now, we can find the conditional probability:

\[ P(\text{Movies} | 31–35) = \frac{\text{Number of people aged 31–35 who like Movies}}{\text{Total number of people aged 31–35}} = \frac{21}{50} \]

Now, we convert this to a percentage:

\[ P(\text{Movies} | 31–35) = \frac{21}{50} \times 100% \]

Calculating this gives:

\[ \frac{21}{50} = 0.42 \] \[ 0.42 \times 100% = 42% \]

Therefore, the probability that a person spends their time going to the movies, given that they are 31–35 years old, is:

\[ \boxed{42%} \]