5.3 Exercise 3

Five world class sprinters are entered in a 200 meter dash. Each has a good chance to break the current track record. There is a thirty percent chance a late cold front will move in, bringing conditions that adversely affect the runners. Otherwise, conditions are expected to be favorable for an outstanding race. Their respective probabilities of breaking the record are:

1. Good weather (no front): 0.75, 0.80, 0.65, 0.70, 0.85
2. Poor weather (front in): 0.60, 0.65, 0.50, 0.55, 0.70

The performances are (conditionally) independent, given good weather, and also, given poor weather. What is the probability that three or more will break the track record?

Hint. If B₃ is the event of three or more, $P \left(B_{3}\right) = P \left(\right. B_{3} \left|\right. W \left.\right) P \left(W\right) + P \left(\right. B_{3} \left|\right. W^{c} \left.\right) P \left(W^{c}\right)$.