Question

let a and b be two events with the following probabilities.

the probability that a occurs is 0.6.

the probability that b occurs is 0.9.

the probability that b occurs given that a occurs is 0.3.

what is the conditional probability that a occurs given that b occurs?

write your answer as a decimal. do not round.

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is $P(B|A)=\frac{P(A\cap B)}{P(A)}$ and $P(A|B)=\frac{P(A\cap B)}{P(B)}$. First, find $P(A\cap B)$ using $P(B|A)$.
We know that $P(B|A) = 0.3$ and $P(A)=0.6$. From $P(B|A)=\frac{P(A\cap B)}{P(A)}$, we can solve for $P(A\cap B)$.
$P(A\cap B)=P(B|A)\times P(A)$.
$P(A\cap B)=0.3\times0.6 = 0.18$.

Step2: Calculate $P(A|B)$

Now that we have $P(A\cap B) = 0.18$ and $P(B)=0.9$, we use the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$.
$P(A|B)=\frac{0.18}{0.9}=0.2$.

Answer:

$0.2$