Question

for two events A and B, P(A) = .4, P(B) = .2, and P(A/B) = .6, find: a- P(A and B) and find b-P(B/A

Answer 1

Conditional Probability : The probability of one event occurring with some relationship to one or more other events.

$\begin{gathered} It\text{ express as :} \\ P(B|A)=\frac{P(A\text{ and B)}}{P(A)} \end{gathered}$

In the given question we have :

P(A) = .4, P(B) = .2 and P(A/B) = 0.6

a) P(A and B)

Simplify the general expression of conditional probability for P(A and B)

$\begin{gathered} P(A|B)=\frac{P(A\text{ and B)}}{P(B)} \\ P(A\text{ and B)=P(B) P(A/B)} \end{gathered}$

Substitute the value in the given expression :

$\begin{gathered} P(A\text{ and B)=P(B) P(A/B)} \\ P(A\text{ and B)=(0.2)(}0.6) \\ P(A\text{ and B)=}0.12 \end{gathered}$

P( A and B ) = 0.12

b) P(B/A)

Now again simplify the general expression for P(B/A)

$P(B|A)=\frac{P(A\text{ and B)}}{P(A)}$

Substitute the value and simplify :

$\begin{gathered} P(B|A)=\frac{P(A\text{ and B)}}{P(A)} \\ P(B|A)=\frac{0.12}{0.4} \\ P(B|A)=0.3 \end{gathered}$

P(B|A) = 0.3

Answer : P( A and B ) = 0.12

P(B|A) = 0.3