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I'm learning probability and it's been going fine. But some angst I can't seem to shake, is how algebra reflects human intuition/understanding and general physical processes. Let me explain.

Consider Blitzstein, Introduction to Probability (2019 2 edn), Chapter 2, Exercise 50, p 94.

Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two” rule: the first player to win two games more than his opponent wins the match. Find the probability that Calvin wins the match (in terms of p).

The answer is simple:

Let $X$ be a random variable that denotes the number of games that Calvin wins out of two games. By LOTP, we have \begin{align*} P(W) &= P(W|X=0)P(X=0) + P(W|X=1)P(X=1) + P(W|X=2)P(X=2) \\ &= 0 + P(W)2pq + p^2 \\ &= \frac{p^2}{1-2pq} \\ &= \frac{p^2}{p^2+q^2} \end{align*}

Everything makes enough sense but something that bothers me is $P(W|X=1)$ = $P(W)$. Intuitively, this makes perfect sense, since losing a game and then winning (or vice versa) is equivalent to just "restarting the game"; this is me speaking intuitively, as a human. But how do I know that the algebra itself reflects this understanding that I as a human am understanding?

My only answer is: it just does. Someone has proved this stuff works.

Also, I think something that's confusing me is the fact that we're defining $P(W)$ in terms of itself, which makes me think it's recursive, but we don't solve it recursively; we solve it algebraically. What I mean is that when I see recursion, I automatically think proof by induction or some type of recurrence, but instead we just do very simple algebraic manipulation. I can't help but wonder whether behind the algebra there's some series that we're converging to and the answer is $P(W)$. Put differently, this simple algebra is actually hiding recursion. But if this is the case, that's so weird because the algebra is completely hiding what's "really going on," which further entrenches in my mind that algebra is really strange.

Moreover, using algebraic manipulation here is strange. Why? Because the $=$ sign is deceiving: we're defining $P(W)$; what seems more appropriate is $P(W):= whatever$ and I'm pretty sure we can't just treat it as $=$ as I've done before, which is required for algebraic manipulation.

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    $\begingroup$ These are really good questions. Have you studied probability from a measure theoretic perspective? That won't clear everything up you're asking here, but it will partially scratch that itch. $\endgroup$ Oct 27, 2021 at 7:45
  • $\begingroup$ How "the algebra itself reflects this understanding"? It does not... we humans use concepts to describe the world: family, marriage, etc. With mathematics it is basically the same: we use abstract mathematical concepts and theories to describe the "facts" of the world. $\endgroup$ Oct 27, 2021 at 8:06
  • $\begingroup$ @DerekAllums I am not. I am in high school still. $\endgroup$
    – beginner
    Oct 27, 2021 at 8:12
  • $\begingroup$ @MauroALLEGRANZA Yes, that makes sense. But at the level I'm currently at, explanations are given in terms of intuition as opposed to rigorous mathematics: my teacher said $P(W|X=1) = P(W)$ because of the loose answer I gave. And this doesn't satisfy me ... $\endgroup$
    – beginner
    Oct 27, 2021 at 8:13
  • $\begingroup$ See Conditional probability: $P(W|X=1)$ is the probability of "event" $W$ happening when we know that event $X=1$ has happened. In general, it is not the same as $P(W)$. "$P(A|B)$ may or may not be equal to $P(A)$ (the unconditional probability of $A$). If $P(A|B) = P(A)$, then events $A$ and $B$ are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other." $\endgroup$ Oct 27, 2021 at 8:19

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