A Middle Manager Is an Avid Runner and Keeps an Informal

Essay

A middle manager is an avid runner and keeps an informal diary of her daily 5-mile training runs. Most of the time, she spends 10 minutes or more stretching before running, believing that this will help prevent minor muscle injuries. In fact, she estimates that 64% of days over the last year, she has stretched and avoided any muscle problems, that is, the relevant joint probability is Pr(Stretch & Healthy) = .64.
(a) Is she correct in concluding that there is a positive association between stretching and being injury free? Now suppose she does some additional thinking and recalls many days when she hadn't bothered to stretch and fortunately still avoided any muscle pulls. Her guess is that Pr(No Stretch & Healthy) = .20. Determine Pr(Stretch|Healthy). Does this indicate that stretching reduces the risk of injury?
(b) Finally, the manager recalls a small number of instances when she neglected to stretch and, indeed, suffered muscle pulls, leading to the guesstimate: Pr(No Stretch and Muscle Pull) = .02. Write down the appropriate two-by-two joint probability table. Does stretching reduce the risk of injury?