For a particular soccer team, given that they win a match, the probability that Amelia is playing is 80%. It is known that Amelia plays in 70% of the games. What is the probability of the soccer team winning a match?
I approached this question by using the conditional probability formula:
$P(A|B) = \frac{P(A∩B)}{P(B)}$
Let A be Amelia playing and M be winning a match
$P(A|M) = \frac{P(A∩M)}{P(M)}$ = $0.8$
To find $P(M)$, I need to know P(A∩M) and then I'm stuck. I thought about using the formula
$P(M|A) = \frac{P(A∩M)}{P(A)}$ where $P(A)$ = $0.7$ but then I didn't know $P(M|A)$.
I also considered using the complements of events but I didn't know where to go with that.
Does anyone have any suggestions?