In a coding contest, there are 3 problems to solve. If the probability of solving each problem is 0.7, 0.8, and 0.9 respectively, what is the probability of solving at least one problem?

Example Answer

If solving the three problems is mutually independent, I would take the complement of solving none. The failure probabilities are 0.3, 0.2, and 0.1, so the probability of no solutions is 0.3 × 0.2 × 0.1 = 0.006. That makes the probability of at least one solution 1 − 0.006 = 0.994, or 99.4%.

Independence is an additional assumption. The three individual probabilities do not tell us how the outcomes overlap. Shared skills or a common time limit could make them dependent, so without information about that dependence there is no unique answer.

Follow-up to prepare for

With only the supplied marginal probabilities, the probability of at least one solution lies between 0.9 and 1. The lower bound follows because solving the third problem already has probability 0.9; the upper bound is 1. Both bounds are attainable. Multiplying failure probabilities requires mutual independence, not merely pairwise independence.