Dating Visually
Matt Gattis tweeted a quiz earlier tonight: 10 girls and 10 guys in a group. Sally dated 5 of the guys, Bob dated 2 of the girls. What’s the probability that Bob dated Sally? Think about it for a bit, then read on.
Kui Tang has a nice write up of the solution over on his blog, but I thought I’d bang out a quick alternate explanation for those of us who like to visualize our probabilities: imagine a 10×10 grid of cells, the x axis corresponding to the men and the y axis to the women, with each cell either on or off depending on whether the x,y pair had been on a date. Take and count up all unique grid configurations that have Sally going on 5 dates and Bob going on 2. That’s your denominator. Your numerator is then the number of these unique grids that have Sally matched with Bob. These are huge numbers, but then recognize that all possible non-Bob/non-Sally cell state configurations repeat for every unique Bob/Sally configuration, and so neatly cancel out.
The math given in Kui’s post is the same thing expressed with counting formulas, but I think picturing the problem as a set of unique grid layouts helps give a better intuitive understanding of what’s going on. It’s hard to accidentally overcount, for instance, because its clear that the visual equivalent to (10 2) * (10 5) counts the Bob+Sally cell too many times, and it erases the questions about the cases where only one other woman has dated at all or 2 other women have dated 10 guys, because it’s clear they’ve been taken into account as part of the massive number of states that cancel out when you do the tally.