I know this problem is on Math SE, but my question has nothing to do with the mathematics itself but rather the intuition/"feel" for the problem. Specifically, my question is on part (b) and (c).
You are going to play $2$ games of chess with an opponent whom you have never played against before (for the sake of this problem). Your opponent is equally likely to be a beginner, intermediate, or a master. Depending on which, your chances of winning an individual game are $90$%, $50$%, or $30$%, respectively.
(a) What is your probability of winning the first game?
Let $W_1$ be the event "win first game"; let $B$ be the event "chose beginner"; same for $I,M$. We want to find $P(W_1)$. We use the law of total probability: $$P(W_1) = P(W_1|B)P(B)+P(W_1|I)P(I)+P(W_1|M)P(M) = \frac{1}{3}(0.9+0.5+0.3) = \frac{17}{30}$$
(b) Congratulations: you won the first game! Given this information, what is the probability that you will also win the second game (assume that, given the skill level of your opponent, the outcomes of the games are independent)?
Let $W_2$ be the event "win second game." We want to find $P(W_2|W_1)$. $$P(W_2|W_1) = \frac{P(W_2\cap W_1)}{P(W_1)}$$ We know the denominator from (a). We want to find the numerator. $$P(W_2\cap W_1) = P(W_2, W_1 | B)P(B) + P(W_2, W_1 | I)P(I) + P(W_2, W_1 | M)P(M) = $$$$ \frac{1}{3}(P(W_2, W_1 | B) + P(W_2, W_1 | I) + P(W_2, W_1 | M)) = \frac{1}{3}(.9^2+.5^2+.3^2) = \frac{23}{60}$$ Thus, we have $$P(W_2|W_1) = \frac{\frac{23}{60}}{\frac{17}{30}} = \frac{23}{34}$$
I know my answer is mathematically correct, but I'm still slightly confused intuitively. We know $P(W_1)=P(W_2)$ because $W_1,W_2$ are independent. If they're independent, the probability of $W_2$ shouldn't change if we know $W_1$ (by definition of independence). In other words, $P(W_2|W_1)=P(W_1)=P(W_2)$. Of course, this isn't true, but I'm wondering where my intuition is faulty.
(c) Explain the distinction between assuming that the outcomes of the games are independent and assuming that they are conditionally independent given the opponent’s skill level. Which of these assumptions seems more reasonable, and why?
I think my confusion on (c) stems from my confusion on (b).