Question

29 multiple choice 1 point
p(b) = 2/3
p(a ∩ b) = 1/6
what will p(a) have to be for a and b to be independent?
1/2
11/12
1/4
5/6

Explanation:

Step1: Recall the formula for independent events

If \(A\) and \(B\) are independent events, then \(P(A\cap B)=P(A)\times P(B)\).

Step2: Solve for \(P(A)\)

We know \(P(B)=\frac{2}{3}\) and \(P(A\cap B)=\frac{1}{6}\). Substitute into the formula \(P(A)=\frac{P(A\cap B)}{P(B)}\).
So \(P(A)=\frac{\frac{1}{6}}{\frac{2}{3}}\).
Using the rule for dividing fractions \(\frac{a}{b}\div\frac{c}{d}=\frac{a\times d}{b\times c}\), we have \(P(A)=\frac{1\times3}{6\times2}\).
Simplify \(\frac{3}{12}=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\) (the third option)