Question

video lesson 5.3 example 4 – finding conditional probabilities
a store surveys customers of different ages. the survey asks whether they would
like to see the store expand its toy department. the results, given as joint
relative frequencies, are shown in the two - way table.

1. what is the probability that a randomly selected

customer whose age is between 10 and 20 would not
like to see the toy department expanded?
0.739

1. what is the probability that a randomly selected customer who would like to see the toy department

expanded is younger than 10?

age (in years)
< 10
10 - 20

20

yes
0.27
0.06
0.23
no
0.09
0.17
0.18

Explanation:

Step1: Identify relevant data for "Yes" and age <10

From the table, for "Yes" and age <10, the joint relative frequency is \( 0.27 \). For "Yes" (all ages), sum the "Yes" row: \( 0.27 + 0.06 + 0.23 = 0.56 \).

Step2: Apply conditional probability formula

Conditional probability \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). Here, \( A \): age <10, \( B \): "Yes". So \( P(\text{age}<10 | \text{Yes}) = \frac{0.27}{0.56} \approx 0.482 \). Wait, but let's recalculate the sum of "Yes" row: \( 0.27 + 0.06 = 0.33 \), \( 0.33 + 0.23 = 0.56 \). Then \( 0.27 / 0.56 \approx 0.4821 \), approximately \( 0.482 \) or \( \frac{27}{56} \approx 0.482 \). Wait, maybe I miscalculated earlier. Let's do it again: "Yes" row: 0.27 (age <10), 0.06 (10 - 20), 0.23 (>20). Sum: 0.27 + 0.06 = 0.33; 0.33 + 0.23 = 0.56. So \( P(\text{age}<10 | \text{Yes}) = \frac{0.27}{0.56} \approx 0.482 \). Wait, but maybe the intended calculation is correct. Wait, the problem is question 5: probability that a customer who would like to see expanded is younger than 10. So numerator: 0.27 (Yes and age <10), denominator: sum of Yes row (0.27 + 0.06 + 0.23 = 0.56). So \( 0.27 / 0.56 \approx 0.482 \). Wait, but let's check the calculation: 0.27 divided by 0.56. 0.27 ÷ 0.56 ≈ 0.4821, which is approximately 0.482 or \( \frac{27}{56} \approx 0.482 \).

Wait, maybe I made a mistake in the sum. Let's add again: 0.27 (age <10, Yes) + 0.06 (10 - 20, Yes) + 0.23 (>20, Yes) = 0.27 + 0.06 is 0.33, plus 0.23 is 0.56. Correct. Then 0.27 / 0.56 = 27/56 ≈ 0.482.

Answer:

\( \approx 0.482 \) (or \( \frac{27}{56} \approx 0.482 \))