Basic Multiplication Rule vs Conditional Probability I'm looking at old probability problems and trying to interpret them in different ways to test my understanding. Given information in this problem: * probability of being windy = .2 * Probability rain on windy day = .3 * Probability rain on non-windy day = .8 Question: What is probability given day is both windy and rainy? I understand that $P(Wind \wedge Rain) = P(W)P(R|W)$ by interpreting the conditional probability as shrinking the outcome space. Since I'm given the probability of it being windy, I thought I could condition on it like so: $ P(WR) = P(WR | W) = P(W|W) P(R|W) = $ above answer Is this interpretation useful or even accurate? It seems to return the same answer but this part of the statement bothers me $P(WR) = P(WR | W)$. Thank you!

"$P(WR) = P(WR | W) = P(W|W) P(R|W) =$ above answer" is not accurate

* The first equality claims to say that _the probability of wind and rain is the probability of wind and rain given wind_. This is not correct as it ignores the possibility of no wind. A better statement might be $P(WR) = P(WR \mid W) P(W)$

* The second equality claims to say that _the probability of wind and rain given wind is the probability of wind given wind multiplied by the probability of rain given wind_. This is correct but is not particularly informative

* The third equality claims to say that _the probability of wind given wind multiplied by the probability of rain given wind is the probability of wind multiplied by the probability of rain given wind_. This is not correct as $P(W \mid W)=1 \
ot = 0.2 = P(W)$




$P(WR) = P(W)P(R\mid W)$ is a basic statement of conditional probability and gives you the answer since you know $P(W)=0.2$ and $P(R\mid W)=0.3$