Probability of Tom being late to work

Question

So I have this problem here.

The probability that Tom is late for work on a rainy day is 0.4 and the probability that he is late for work on a non-rainy day is 0.1.

Can I state that P(Tom being late for work)=P(Late|Rain)+P(Late|No rain)=0.4+0.1=0.5 ?

Answer 1

Assuming that "the probability that tom is late for work on a rainy day" is in reference to $Pr(\text{Late}\mid \text{Rain})$ then no. You need to know the probability of it being a rainy day. The correct statement would be:

$Pr(\text{Late}) = Pr(\text{Rain})Pr(\text{Late}|\text{Rain}) + Pr(\text{No rain})Pr(\text{Late}|\text{No rain})$

This is the law of total probability.

Answer 2

another person has already answered this correctly, but I just wanted to help you think through this. Imagine that everyday is rainy: there would be a 40 percent chance that he is late on a daily basis. Now imagine that it is never rainy: there is a 10 percent chance that he is late. So, having a %50 percent chance of being late is impossible, the answer must be between these two values (0.1-0.4).