Question

use the contingency table below to find the probability of the following events.
a. ( a|b )
b. ( a|b )
c. ( a|b )
d. are events a and b independent?
a. ( p(a|b) = square ) (round to two decimal places as needed.)
contingency table: row b: a=10, a=20; row b: a=20, a=40

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). From the contingency table, we first find the number of elements in \( A \cap B \) and \( B \).

Step2: Identify Frequencies from Table

Looking at the table, for event \( B \), the total number of elements is the sum of the values in the \( B \) row: \( 10 + 20 = 30 \). The number of elements in \( A \cap B \) (the cell where \( A \) and \( B \) intersect) is \( 10 \).

Step3: Calculate Probabilities

First, \( P(A \cap B) = \frac{\text{Frequency of } A \cap B}{\text{Total Number of Observations}} \), and \( P(B) = \frac{\text{Frequency of } B}{\text{Total Number of Observations}} \). The total number of observations is \( 10 + 20 + 20 + 40 = 90 \)? Wait, no, wait. Wait, the table rows are \( B \) and \( B' \), columns are \( A \) and \( A' \). So:

• For \( B \) row: \( A \) is 10, \( A' \) is 20. So total for \( B \): \( 10 + 20 = 30 \).
• For \( B' \) row: \( A \) is 20, \( A' \) is 40. So total for \( B' \): \( 20 + 40 = 60 \).
• Total observations: \( 30 + 60 = 90 \).

Now, \( P(A \cap B) = \frac{10}{90} \), \( P(B) = \frac{30}{90} \).

Then \( P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{10}{90}}{\frac{30}{90}} = \frac{10}{30} = \frac{1}{3} \approx 0.33 \) (rounded to two decimal places).

Wait, let's check again. The formula for conditional probability using frequencies: \( P(A|B) = \frac{n(A \cap B)}{n(B)} \), where \( n(A \cap B) \) is the number of elements in both \( A \) and \( B \), and \( n(B) \) is the number of elements in \( B \).

From the table, \( n(A \cap B) = 10 \) (the cell where row \( B \) and column \( A \) meet), and \( n(B) = 10 + 20 = 30 \) (sum of row \( B \)). So \( P(A|B) = \frac{10}{30} = \frac{1}{3} \approx 0.33 \).

Answer:

\( 0.33 \)