##### Section 3.2 Example 2a Sheldon Ross on Conditional Probability

A student is taking a one-hour-time-limit makeup examination. Suppose the probability that the student will finish the exam in less than x hours is x/2, for all $0 \leq x \leq 1$.Then, given that the student is still working after .75 hour, what is the conditional
probability that the full hour is used?

Let Lx denote the event that the student finishes the exam in less than x hours, $0 \leq x \leq 1$, and let F be the event that the student uses the full hour. Because F is the event that the student is not finished in less than 1 hour

$P(F) = P(L_1^c) = 1 - P(L_1) = 1 - \frac{1}{2} = 0.5$

Now, the event that the student is still working at time .75 is the complement of the event L.75, so the desired probability is obtained from

$P(F \ | \ L_.75^c ) = \frac{P(FL_.75^c)}{P(L_.75^c)} = \frac{P(F)}{1 - P(L_.75)} = \frac{.5}{.625} = .8$

∴ Desired Probability = 0.8