This commuter owns a car, do they take the bus?

Question

Among 300 commuters, 180 take the bus and 120 take the train (each uses just one). Of the 180 bus riders, 60 own a car. Of the 120 train riders, 90 own a car. You pick a commuter at random and they own a car. What is the probability they take the bus?

Accepted Answer

The clue "owns a car" restricts you to car owners, so count them first. - Bus riders who own a car: 60. - Train riders who own a car: 90. Total car owners: 60 + 90 = 150. Of those, the bus riders number 60: P(	ext{bus} \mid 	ext{car owner}) = \frac{60}{150} = \frac{2}{5} = 0.4. Even though bus riders outnumber train riders overall, car ownership is more common among train riders, so among car owners the bus is the minority. Takeaway. Conditioning can flip which group looks bigger; the majority mode overall need not be the majority within the conditioned subgroup.

This commuter owns a car, do they take the bus?

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