A man owns two old cars, A and B, and has trouble starting them on cold mornings. The probability both will start is 0.1; the probability B starts and A does not is 0.1; the probability that neither starts is 0.4.
a) Find the probability that car A will start.
b) Find the probability that car A will start, given car B starts.
c) Find the probability that car B will start, given car A starts.

Please show detailed solution. Thanks!

Draw a rectangle, representing the sample space. The sum of all probabilities in the rectangle equals 1.

Within the rectangle, draw two circles A and B that have a portion of their area overlapping.

There are four distinct non-overlapping areas:

1. The area where A and B overlap, whose probability is .1

2. The remaining portion of B, which is B but not A, and whose probability is also .1

3. The portion of the rectangle outside the two circles, which is neither A nor B, and whose probability is .4

4. The remaining portion of A, which is A but not B, and whose probability is also is 1 - (.1 +.1 +.4) = .4

Using the diagram,

a) areas 1 + 4: .4 + .1 = .5
b) area 1 / (areas 1 + 2): .1/.2 = .5
c) area 1 / (areas 1 + 4): .1/.5 = .2