Question

Your weather app says there's a 30% chance of rain that day. You know that if it rains, the probability that the bus runs late is 40%. If it doesn't rain, the probability that the bus runs late is only 15%. Use proper notation to state the probabilities. (a) What is the probability that it will rain and the bus will be late? (b) What is the probability that the bus will be late? (c) Given that the bus ran late, what was the probability that it was not raining?

Answer 1

Kindly check explanation

Explanation:

Given the following :

P(rain) = p(R) = 30% = 0.3

P(being late Given that it rains) = P(late | rain) = p(L|R) = 0.4

P(being late Given no rain) = P(late | no rain) = 0.15

(a) What is the probability that it will rain and the bus will be late? = P(RAINnLate) = P(RnL)

P(RnL) = p(R) * p(L|R)

P(RnL) = 0.3 * 0.4

= 0.12

(b) What is the probability that the bus will be late?

P(L) = p(R) * p(L|R) + p(no rain) * p(late | no rain)

P(L) = (0.3 * 0.4) + (1 - 0.3)*(0.15)

P(L) = 0.12 + 0.105

P(L) = 0.225

(c) Given that the bus ran late, what was the probability that it was not raining?

Given that bus ran late, the probability that it was not raining = p(no rain | Late)

p(no rain | Late) = 1 - p(R | L)

Recall :

P(RnL) = p(L) * p(R|L)

0.12 = 0.225 * p(R|L)

p(R|L) = 0.12 / 0.225

p(R|L) = 0.5333

p(no rain | Late) = 1 - 0.533

= 0.46666

= 0.467