Insisting with the basics

Probability Level 3

Let's suppose $A$ and $B$ be events in a sample space $\Omega$ , with $P(A) = 0.5$ , $P(A \cup B) = 0.7$ and $P(A|B) = 0.4$ .

Calculate $P(B)$ .

1 solution

Brian Charlesworth
Aug 28, 2016

We have that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$\Longrightarrow P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.5 + P(B) - 0.7 = P(B) - 0.2$ .

Now $P(A|B) = \dfrac{P(A \cap B)}{P(B)} \Longrightarrow P(A \cap B) = P(A|B)*P(B)$ , and so

$P(B) - 0.2 = 0.4*P(B) \Longrightarrow 0.6*P(B) = 0.2 \Longrightarrow P(B) = \boxed{0.333}$ to 3 decimal places.

Exact Brian, thank you, (+1) $\uparrow$

Guillermo Templado - 4 years, 9 months ago