To find the probability of a family having exactly 5 people based on the given survey data, follow these steps:
1. Identify the total number of families surveyed: This is done by summing the frequencies of all family sizes.
[tex]\[
\text{Total number of families} = 87 + 50 + 61 + 31 + 16 + 3 + 2
\][/tex]
[tex]\[
\text{Total number of families} = 250
\][/tex]
2. Identify the number of families with exactly 5 people: From the table, this frequency is given directly.
[tex]\[
\text{Number of families with 5 people} = 31
\][/tex]
3. Calculate the probability: The probability of a family having 5 people, \( P(5) \), is the ratio of the number of families with 5 people to the total number of families surveyed.
[tex]\[
P(5) = \frac{\text{Number of families with 5 people}}{\text{Total number of families}}
\][/tex]
[tex]\[
P(5) = \frac{31}{250}
\][/tex]
4. Simplify the ratio if necessary (here it's already in its simplest form).
5. Convert the simplified ratio into a decimal form if needed for better interpretation of the probability.
[tex]\[
P(5) = \frac{31}{250} = 0.124
\][/tex]
So the probability that a family chosen at random from this survey has 5 people is [tex]\( 0.124 \)[/tex] or [tex]\( 12.4\% \)[/tex].
