For this most recent NBA season, ESPN.com has had in-game win probabilities:

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My question:

Say a team gets a big lead and has a 90% chance of winning. What is the probability that at some point during the rest of the game, the other team stages a comeback to the point where both teams have a 50% chance of winning?

While neither assumption holds in reality, assume for this puzzle that win probabilities are perfectly accurate and continuous (meaning the probabilities don't make a sudden jump, but smoothly transition from one value to another). Surprisingly, I don't believe you need to know anything about the teams, how much time is left, or really anything about basketball, to solve this puzzle.

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    $\begingroup$ How is this exactly a puzzle? This strikes me as no different than how to predict the stock market - you could look up online problems, or similar stock algorithms if you were curious. Mathematically speaking, there's nothing here about the actual weighting of the game, and as you point out nothing holds in reality - in fact, it's just assumptions. What makes the probabilities change? $\endgroup$ Jun 7 '18 at 19:53
  • $\begingroup$ If I don't need to know anything about basketball, how many minutes are remaining, what the maximum point-per-minute rate is, or how dependent on a single player that could become injured, etc, then I would argue you could rewrite this puzzle as just "In a system with exactly two outcomes, if the current probability of outcome A is 90%, what is the probability that the outcome will become 50% within some arbitrary but finite time frame?" which is a meaningless question because of its imprecision and vagueness. $\endgroup$ Jun 7 '18 at 20:22
  • $\begingroup$ I guess the continuity assumption is that for the Cavs to win their win probability must pass through 50% at some point, but as noedne pointed out, this model is not true in reality (say, imagine their only chance of winning is hitting a shot from the other side of the court: that is very unlikely, but if they do it, the probability will jump from 90% to 0% without every going through 50%). $\endgroup$
    – ffao
    Jun 7 '18 at 20:50
  • $\begingroup$ @ffao Actually, the question addresses that by assuming that win probabilities are continuous; in my first model, the probability that both teams' win probabilities become 50% is 10% because of this continuity. $\endgroup$
    – noedne
    Jun 7 '18 at 20:52
  • $\begingroup$ @noedne, your first model violates the accuracy condition: when the probability hits 50%, the chance of a Cavs win is actually 100%. $\endgroup$
    – ffao
    Jun 7 '18 at 20:53

The only way to answer this question is to use both assumptions, which the question itself admits are not entirely realistic.

Since the function is continuous,

For every Cavs win, the end value of the probability the other team wins is 0%, and it started at 90%. That means that by the intermediate value theorem it must go through 50% at some point.

Furthermore, because it is perfectly accurate,

Of the games the probability function goes to 50% on, exactly half of those are wins. Since we determined all wins must hit 50%, this means 50% is hit twice as much as there are wins, or 20% of the time.

  • $\begingroup$ Woah that was a nice one, I was going the right way and had the same conclusion as your first paragraph, but was off in the second one. That's smart $\endgroup$ Jun 8 '18 at 13:27
  • $\begingroup$ Not positive this is correct but still it sounds logical. +1 $\endgroup$
    – paparazzo
    Jun 8 '18 at 23:57

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