Marksman Probability Puzzle

Bruce Ediger

Found a puzzle on the notorious Futility Closet web site.

Here’s Futility Closet’s puzzle. Mr or Ms Closet decided to call the puzzle “Gun Control”. I don’t know if they chose the title to skirt controversy or something.

Problem Statement

Marksman A hits a certain small target 75 percent of the time. Marksman B hits it 25 percent of the time. The two of them aim at that target and fire simultaneously. One bullet hits it. What’s the probability that it came from A?

I decided to solve this via a simulation.

Simulation

  1. Chose a random number between 0 and 100 inclusive. If that random number evaluates to less than 75, Marksman A hit the target.
  2. Chose a random number between 0 and 100 inclusive. If that random number evaluates to less than 25, Marksman b hit the target.
  3. If A or B (but not both!) hit the target, count who hit the target, A or B
  4. Repeat a few times, display percentages of who hit the target.

Code repo

Results of Simulation

The simulation program worked reasonably well when I had it do 10,000 trials, but it seemed to work better at 100,000 trials.

% ./gc1 -N 100000
A 0.901
B 0.099
0: 18777 (0.188)        1: 62422 (0.624)        2: 18801 (0.188)

I simulated 90.1% of the times that one bullet hit the target, Marksman A shot it. The Futility Closet answer is 90%

Yes, but why?

The question is: “why 90% ?”

I think I understand the 188% where both A and B hit the target. A hits the target 75% of the rounds. B hits the target 25% of the time, and that’s independent of A’s marksmanship. So 0.25*0.75 = .1875. The probability of neither A nor B hitting the target works out to the same number, but backwards: A misses 25% of the time, B misses 75% of that, 0.75*0.25 = .1875.

That leaves 1.00 - 0.1875 - 0.1875 = 0.625, 62.5% of the time, only one of A and B hits the target.

For 18.75% of the rounds, both A and B hit the target.

For marksman B: 25 - 18.75 = 6.25%. Marksman B hits the target when A does not. 6.25% is 10% of the 62.5% of the rounds that only one marksman hits the target. That leaves Marksman A as the only one hitting the target 90% of the rounds when only one marksman hits the target.

This is a completely unintuitive result to me. Probability is hard.