This week the question is: Can You Make an Incredible Comeback?
Suppose you’re playing a game in which there are five “possessions.” For each possession, there’s a 50 percent chance that your team scores one point. If you don’t score, then your opponent instead scores one point.
After the game, ESPN reports that your opponent’s chances of winning were “75 percent chance or higher” at some point during the game (i.e., before the final possession is complete).
Given this information, what was the probability that your team actually won the game?
Highlight to reveal solution:
First I calculate score specific probabilities. Like, if I need to win 1 more possession, and there are 2 possessions left. What’s the probability, I will succeed? Or, if I need to win 2 more possessions, and there are 4 possessions left. What’s the probability I will succeed? In the latter case, I add up the probabilities of getting 2, 3 or 4 wins. So that’s sheet 1.
As the next step I look at the possible scores, before the final possession: 0-0, 1-0, 2-0 etc. Let’s use 2-0 as an example. With that score, I need 1 more won possession, and there are 3 possessions left. I need 1 with 3 left, I calculated that above, as 87.5% chance. That’s sheet 2.
For the final step, I look at all possible games (there are only 32). I mark each “team 1 had a 75% or more chance of winning” state, like 2-0 mentioned above. I mark whether team 2 ended up winning. I count. In 3 out of 16 cases, team 1 was close to winning, but didn’t. 3/16 = 18.75%. And that’s sheet 3.
Stuff from ommadawn.dk (Lise Andreasen) 
