Chance a commuter uses the bus or the train

Question

In a city survey, 70\% of workers use the bus during a typical week, 45\% use the train, and 25\% use both. What is the probability that a randomly chosen worker uses at least one of the two?

Accepted Answer

"At least one" means bus or train or both. Adding 70\% and 45\% counts the both-users twice, once in each group. Inclusion-exclusion subtracts the overlap once: P(	ext{bus or train}) = 0.70 + 0.45 - 0.25 = 0.90. So 90\% use at least one. As a check, the disjoint pieces are: only bus 0.45, only train 0.20, both 0.25, neither 0.10, which sum to 1.00. Takeaway. The union of two overlapping groups is the sum of their sizes minus their shared part.

Chance a commuter uses the bus or the train

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