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The distribution of pairs of shoes in some teenagers' closets is as follows:

\begin{tabular}{|c|c|c|c|c|c|}
\hline
Pairs of Shoes & 1 & 2 & 3 & 4 & 5 \\
\hline
Frequency & 18 & 30 & 57 & 30 & 15 \\
\hline
\end{tabular}

Find the probability that a teenager has exactly 4 pairs of shoes in their closet.

[tex]\[ P(4) = [?] \][/tex]


Sagot :

To determine the probability that a teenager has exactly 4 pairs of shoes in their closet, follow these steps:

1. Gather the data:
- The pairs of shoes and their respective frequencies are given.
- Pairs of shoes: [tex]\(1, 2, 3, 4, 5\)[/tex]
- Frequency: [tex]\(18, 30, 57, 30, 15\)[/tex]

2. Calculate the total number of teenagers:
- Add up all the frequencies to find the total number of teenagers surveyed.
[tex]\[ \text{Total number of teenagers} = 18 + 30 + 57 + 30 + 15 = 150 \][/tex]

3. Identify the frequency of teenagers with exactly 4 pairs of shoes:
- From the frequency table, we see that the frequency for teenagers with 4 pairs of shoes is [tex]\(30\)[/tex].

4. Calculate the probability:
- Probability [tex]\(P(4)\)[/tex] is given by dividing the number of teenagers with exactly 4 pairs of shoes by the total number of teenagers.
[tex]\[ P(4) = \frac{\text{Number of teenagers with 4 pairs of shoes}}{\text{Total number of teenagers}} = \frac{30}{150} \][/tex]

5. Simplify the fraction:
- Divide both numerator and denominator by their greatest common divisor (GCD), which is 30 in this case.
[tex]\[ P(4) = \frac{30}{150} = \frac{1}{5} = 0.2 \][/tex]

Therefore, the probability that a teenager has exactly 4 pairs of shoes in their closet is:

[tex]\[ P(4) = 0.2 \][/tex]