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?