Can You Solve This?

Added: 2026-05-09

[00:00:00]Here's a problem from Frederick Mosteller's 50 challenging problems in probability that will completely flip your understanding of optimal strategy.

[00:00:08]Three sharpshooters, A, B, and C, face off in a pistol duel at high noon.

[00:00:13]Everyone knows the exact shooting abilities. A hits his target 30% of the time. B never misses, 100% accuracy. C hits his target 50% of the time. They'll fire in order, A first, then B, then C, repeating this cycle. If someone gets hit, they die instantly and are out of the game. The survivors continue taking turns until only one remains. The question, what should A's strategy be?

[00:00:39]Who should A aim at? If A try to eliminate the middle threat, C, with his 50% accuracy and successfully hits him, what happens next? It's now B's turn to shoot, and there are only two people left, A and B. Since B never misses, this scenario has only one possible outcome. B kills A immediately. A gains absolutely nothing by shooting at C, because it guarantees his own death on the very next turn. Even if A misses C, he's wasted his turn without improving his position. Shooting at C is strategically pointless. With C ruled out, conventional wisdom says A should aim at B, the perfect marksman who poses the greatest threat. Let's analyze what happens in each scenario when A shoots at B. Scenario one, A shoots at B and hits, 30% chance. If A successfully eliminates B, the duel becomes a one-on-one between A and C. But here's the catch, C gets the next shot. This is where the mathematics get fascinating.

[00:01:37]A's survival depends on this deadly back-and-forth. C shoots first, 50% chance to hit A. If C misses, 50% chance. A gets to shoot back, 30% chance to hit C. If A misses, too, 70% chance.

[00:01:53]We're back to C shooting again. Let's calculate A's survival probability step by step. First exchange possibilities. C misses, A hits, 0.5 * 0.3 = 0.15 * C misses, A misses, 0.5 * 0.7 = 0.35.

[00:02:11]Second exchange, if both missed first time. C misses, A hits, 0.35 * 0.5 * 0.3 = 0.0525.

[00:02:22]C misses, A misses, 0.35 * 0.5 * 0.7 = 0.1225.

[00:02:30]Notice the pattern. Each complete miss-miss cycle multiplies our probability by 0.5 * 0.7 = 0.35. This creates what mathematicians call a geometric series. A's total survival probability = 0.15 + 0.0525 + 0.018375 and so on. In general form, 0.15 + 0.15 * 0.35 + 0.15 * 0.35 squared + 0.15 * 0. 35 cubed and so on. This is a geometric series with first term, A = 0.15, common ratio, R = 0.35. A geometric series is an infinite sum where each term is the previous term multiplied by a constant ratio. The beauty is that when R's are less than one, which it is here, since 0.35 less than one, the infinite sum converges to a finite value. The formula for the sum of an infinite geometric series is A divided by 1 - R. Plugging in our values, A's survival probability = 0.15 / 1 - 0.35 = 0.15 / 0.65 = 3/13, approximately 0.23 or 23%. Scenario two, A shoots at B and misses, 70% chance. When A misses B, it's B's turn to shoot. Now B faces a choice. Shoot the weak A, 30% accuracy, or the moderate threat C, 50% accuracy.

[00:04:13]Considering B is logical and strategic, since C poses a significantly greater threat, B will eliminate C first. With C gone, it's just A versus B, and A gets exactly one chance to hit B before B's perfect aim ends the duel. A's survival probability in this case, 30%, just his shooting accuracy. Combining both scenarios, 30% chance A hits B leads to 23% survival rate. 70% chance A misses B leads to 30% survival rate. A's expected survival aiming at B becomes 0.3 * 0.23 + 0.7 * 0.3 = 0.069 + 0.21 = 0.279 or 27.9%.

[00:05:02]But wait, what if A deliberately misses his first shot? Purposely shoots at the ground. If A intentionally misses, B gets to shoot next. B will logically eliminate C, the bigger threat, leaving just A and B. But crucially, A now gets the first shot in this final duel. A's survival probability becomes 30%, which is his shooting accuracy. A survives more often by deliberately missing, 30%, than by trying to hit B, 27.9%.

[00:05:31]Why this strategy works. A is essentially using B as a weapon against C. A recognizes that, one, he can't eliminate both opponents. Two, B will eliminate C regardless of what A does.

[00:05:43]Three, the key is positioning himself optimally for the final one-on-one. By doing nothing, A ensures he gets the first shot against B in their final confrontation, a better position than facing either opponent who gets to shoot first. This problem reveals how geometric series appear in probability calculations involving repeated trials.

[00:06:02]Each miss-miss cycle in the A versus C duel represents another term in our infinite series. Understanding geometric series is crucial because they model many real-world scenarios, the probability of eventual success in repeated attempts, population growth models, financial calculations involving compound effect, any situation where outcomes multiply through repeated processes. The convergence to 3/13 means that no matter how long A and C might theoretically duel, A's survival probability remains exactly 23.08%.

[00:06:35]This strategic inaction principle appears everywhere. In negotiations, sometimes the best move is letting other parties argue while you stay neutral. In competitive markets, smaller companies often benefit when larger competitors fight each other. This duel problem proves that sometimes the best action is no action at all. Our instincts scream, "Shoot the biggest threat." But mathematics whispers, "Let others do your work for you." The geometric series calculation reveals the precise probabilities that our gut feelings completely misunderstand. What seems like cowardice is actually the mathematically optimal strategy. Here's a final question. If A's accuracy improved to 40%, would the optimal strategy change? Think about it. Better accuracy means A has a better chance in any final duel, making the deliberate miss strategy even more attractive since it guarantees A gets the first shot against B. Mathematics might teach us to act against our instincts, but they never lie about the optimal path to survival.