What is the conceptual difference between the probability of winning two successive games in a playoff series vs winning the second game given that you won the first game?
I understand P(W2|W1) as the probability of winning the second game given you've won the first game, but I can't see how to interpret P(W2, W1), or if it even applies in this context; I think to get to W2 you have to have W1 as a given.
I can find similar examples here and here but they lack the successive nature of a series of games.
I've mapped out my cases as tree of possibilities to try to see how P(W2, W1) would fit in. Is there a better way to think about this?
$\mathsf P(W_1,W_2)$ is the joint probability of the events; that is the probability of their intersection, also written as $\mathsf P(W_1\cap W_2)$. The probability that both events have occurred (win game 1 and game 2).
$\mathsf P(W_2\mid W_1)$ is the conditional probability that $W_2$ occurs when it is given that $W_1$ has.
By the definition for conditional probability, we have that:
$$\mathsf P(W_2\mid W_1)=\dfrac{\mathsf P(W_1, W_2)}{\mathsf P(W_1)}$$
Or $\mathsf P(W_1,W_2)=\mathsf P(W_1)\cdot \mathsf P(W_2\mid W_1)$