A win two matches from now

Question

A football club either wins a match or does not win it, and its next result depends only on the current one: - After a win, the next match is a win with probability 0.5 (and not a win with probability 0.5). - After a match that is not a win, the next is a win with probability 0.25 (and not a win with probability 0.75). The club just won today's match. What is the probability that it wins the match two rounds from now?

Accepted Answer

Two rounds ahead is two steps. We branch on the result of the match in between and keep the routes that end in a win. We start from a win, the middle match can go either way, and the last match must be a win. - Win, then win, then win: 0.5 	imes 0.5 = 0.25. - Win, then not a win, then win: 0.5 	imes 0.25 = 0.125. Add the two winning routes: 0.25 + 0.125 = 0.375. So the club wins the match two rounds out with probability 0.375, or 	frac{3}{8}. Takeaway. Chaining two steps means splitting on the middle state, multiplying the one-step chances along each path, and adding the paths that finish …

A win two matches from now

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