Daily Math Puzzle: 2026-03-31

75% — Let A be the event that the Alpha atom decays, and B be the event that the Beta atom decays. We are given: - P(A) = 0.50 (Alpha decays) - P(not A) = 1 - 0.50 = 0.50 (Alpha does not decay) - P(B) = 0.25 (Beta decays) - P(not B) = 1 - 0.25 = 0.75 (Beta does not decay) The atoms decay independently. First, let's list all possible outcomes after one hour and their probabilities: 1. Both Alpha and Beta decay: P(A and B) = P(A) * P(B) = 0.50 * 0.25 = 0.125 2. Alpha decays, Beta does not decay: P(A and not B) = P(A) * P(not B) = 0.50 * 0.75 = 0.375 3. Alpha does not decay, Beta decays: P(not A and B) = P(not A) * P(B) = 0.50 * 0.25 = 0.125 4. Neither Alpha nor Beta decays: P(not A and not B) = P(not A) * P(not B) = 0.50 * 0.75 = 0.375 (Note: The sum of these probabilities is 0.125 + 0.375 + 0.125 + 0.375 = 1.00) We are given the condition that "exactly one of the two atoms has decayed." Let's call this event E. Event E occurs in two ways: - Alpha decays AND Beta does not decay (outcome 2 above). - Alpha does not decay AND Beta decays (outcome 3 above). The probability of event E is the sum of these probabilities: P(E) = P(A and not B) + P(not A and B) = 0.375 + 0.125 = 0.50. Now, we want to find the probability that the decayed atom was Alpha, *given* that exactly one atom decayed. This is a conditional probability: P(Alpha decayed | E). The event "Alpha decayed AND exactly one atom decayed" means that Alpha decayed and Beta did not decay (because if Beta also decayed, it wouldn't be 'exactly one'). This is exactly outcome 2: (A and not B), which has a probability of 0.375. Using the formula for conditional probability: P(X|Y) = P(X and Y) / P(Y) P(Alpha decayed | E) = P(A and not B) / P(E) = 0.375 / 0.50 = 0.75 So, the probability that the decayed atom was the Alpha atom, given that exactly one atom decayed, is 75%.