Question

It's an overcast day in Hayward and you're waiting for AC Transit 97 Bus Line to show up. 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

a

`P(r n L) =  0.12`

b

`P(L) = 0.225`

c

`P(\= r | L) = 0.4667`

Explanation:

The chance that it will rain is
`P(r ) = 0.30`

The chance that it does not rain is
`P(\= r ) = 1 - P(r )`

`P(\= r ) = 1 -0.30`

`P(\= r ) = 0.70`

The probability that the bus run late if it rains is
`P(L | r) = 0.4`

The probability that the bus run late if it does not rain is
`P(L | \= r) = 0.15`

Generally the probability that it will rain and the bus will be late is mathematically represented as

`P(r n L) = P(L | r) * P(r)`

=>
`P(r n L) = 0.4 * 0.30`

=>
`P(r n L) = 0.12`

Generally the probability that the bus will be late is mathematically represented as

`P(L) = P(r) * P(L|r) + P(L | \= r) * P(\= r)`

=>
`P(L) = 0.30 * 0.4 +0.15*0.70`

=>
`P(L) = 0.225`

Generally given that the bus ran late, the probability that the bus it was not raining is mathematically represented as

`P(\= r | L) = 1- P(r | L )`

Here
`P(r | L ) = (P(r n L))/(P(L))`

=>
`P(r | L ) = (0.12)/(0.225)`

=>
`P(r | L ) = 0.533`

So

`P(\= r | L) =  1- 0.533`

=>
`P(\= r | L) =  0.467`