Probability Question

John is a very forgetful person, especially when it comes to his umbrella. Suppose that in the town where John lives, the probability that it is raining on any given day is
0.6. Furthermore, suppose that on 40% of all days, it is both raining and John has forgotten his umbrella at home. Calculate the probability that on a rainy day, John
will forget his umbrella at home.

What I have so far is that P(rain) = 0.6 and P(rain ∩ forgot umbrella) = 0.4. So how would I calculate P(forgot umbrella | rain)?

I'm also not quite sure how to do the following two questions:

Suppose John realizes that given it is not raining, he has not forgotten his umbrella
80% of the time. Find the probability that, regardless of the weather, John forgets his
umbrella.

On his way to work, John realizes that he has forgotten his umbrella. Knowing this,
what is the probability that it is raining outside?

Thanks in advance!

Mashiro said:

John is a very forgetful person, especially when it comes to his umbrella. Suppose that in the town where John lives, the probability that it is raining on any given day is 0.6. Furthermore, suppose that on 40% of all days, it is both raining and John has forgotten his umbrella at home. Calculate the probability that on a rainy day, John will forget his umbrella at home.

What I have so far is that P(rain) = 0.6 and P(rain ∩ forgot umbrella) = 0.4. So how would I calculate P(forgot umbrella | rain)?

Click to expand...

\(\displaystyle \mathcal{P}(U|R)=\dfrac{\mathcal{P}(U\cap R)}{\mathcal{P}(R)}\)

Imagine 100 days. "the probability that it is raining on any given day is 0.6." So it is raining on 60 of those days. " on 40% of all days, it is both raining and John has forgotten his umbrella at home." So on 40 of those days it is raining and John forgot his umbrella.

"Calculate the probability that on a rainy day, John will forget his umbrella at home." Out of 60 rainy days, John forgot his umbrella 40 days. Probability that John will forget his umbrella on a rainy day, 40/60= 2/3.