Question

1. p(a) = 1/3. p(b) = 1/2, p(a ∪ b) = 1/6

d) calculate p(a ∩ b)
e) calculate p(b|a)
f) are the events independent? give a reason:

1. create a diagram for the shaded region of:

c) p(a ∪ b)
d) complement of the probability in (b)

Explanation:

Step1: Use the formula for \(P(A\cup B)\)

The formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Rearranging for \(P(A\cap B)\) gives \(P(A\cap B)=P(A)+P(B)-P(A\cup B)\).
Substitute \(P(A)=\frac{1}{3}\), \(P(B)=\frac{1}{2}\), \(P(A\cup B)=\frac{1}{6}\) into the formula:
\(P(A\cap B)=\frac{1}{3}+\frac{1}{2}-\frac{1}{6}\)

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

Find a common denominator (6).
\(\frac{1}{3}+\frac{1}{2}-\frac{1}{6}=\frac{2 + 3-1}{6}=\frac{4}{6}=\frac{2}{3}\)

Step3: Use the formula for conditional probability \(P(B|A)\)

The formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\)
Substitute \(P(A\cap B)=\frac{2}{3}\) and \(P(A)=\frac{1}{3}\)
\(P(B|A)=\frac{\frac{2}{3}}{\frac{1}{3}} = 2\)

Step4: Check for independence

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\)
\(P(A)\times P(B)=\frac{1}{3}\times\frac{1}{2}=\frac{1}{6}
eq\frac{2}{3}=P(A\cap B)\)

Answer:

d) \(P(A\cap B)=\frac{2}{3}\)
e) \(P(B|A) = 2\)
f) The events are not independent. Because \(P(A\cap B)
eq P(A)\times P(B)\)