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]
