On any day in a city, the weather is sunny with probability 0.5, cloudy with probability 0.3 and rainy with probability 0.2. The probability of a traffic jam on the main road is 0.1 on a sunny day, 0.2 on a cloudy day and 0.5 on a rainy day. The probability that there is a traffic jam on the main road on a day chosen at random is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 0.21
Let \(E_1\), \(E_2\), \(E_3\) be the events that the day is sunny, cloudy, rainy, and J the event of a traffic jam. Exactly one kind of weather happens on a day, so these events form a partition.
\(P(E_1) = 0.5,\quad P(E_2) = 0.3,\quad P(E_3) = 0.2\)
\(P(J|E_1) = 0.1,\quad P(J|E_2) = 0.2,\quad P(J|E_3) = 0.5\)
By the theorem of total probability:
\(P(J) = P(E_1)P(J|E_1)+P(E_2)P(J|E_2)+P(E_3)P(J|E_3)\)
\(P(J) = 0.5\times 0.1+0.3\times 0.2+0.2\times 0.5\)
\(P(J) = 0.05+0.06+0.10 = 0.21\)