A survey found the distribution of some families by size, as follows:

\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Family Size & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
Frequency & 87 & 50 & 61 & 31 & 16 & 3 & 2 \\
\hline
\end{tabular}

Find the probability of a family having 5 people.

[tex]\[ P(5) = [?] \][/tex]



Answer :

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].

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