Question

given the following venn diagram:
data

note: when answering the questions below, for any final answer that has up to four decimal
places, enter your answer without rounding the number. for any answers with more than four
decimal values, round your final answer to four decimal places.

a) calculate the probability ( p(a|b) =? )

b) calculate the probability ( p(b|a) =? )

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \( P(A|B)=\frac{P(A\cap B)}{P(B)} \) and \( P(B|A)=\frac{P(A\cap B)}{P(A)} \).

Step2: Calculate \( P(A) \) and \( P(B) \)

\( P(A)=0.29 + 0.18=0.47 \), \( P(B)=0.45+0.18 = 0.63 \), and \( P(A\cap B)=0.18 \).

Step3: Calculate \( P(A|B) \)

Substitute into the formula \( P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{0.18}{0.63}=\frac{2}{7}\approx0.2857 \).

Step4: Calculate \( P(B|A) \)

Substitute into the formula \( P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{0.18}{0.47}=\frac{18}{47}\approx0.3830 \).

Answer:

a) \( 0.2857 \)
b) \( 0.3830 \)