5.3 Exercise 5 — Applied Probability

Seven students plan to complete a term paper over the Thanksgiving recess. They work independently; however, the likelihood of completion depends upon the weather. If the weather is very pleasant, they are more likely to engage in outdoor activities and put off work on the paper. Let E_i be the event the ith student completes his or her paper, A_k be the event that k or more complete during the recess, and W be the event the weather is highly conducive to outdoor activity. It is reasonable to suppose $\left\{E_{i} : 1 \leq i \leq 7\right\} ci \left|\right. W$ and $ci \left|\right. W^{c}$ . Suppose

P \left(\right. E_{i} \left|\right. W \left.\right) = 0 . 4 , 0 . 5 , 0 . 3 , 0 . 7 , 0 . 5 , 0 . 6 , 0 . 2

P \left(\right. E_{i} \left|\right. W^{c} \left.\right) = 0 . 7 , 0 . 8 , 0 . 5 , 0 . 9 , 0 . 7 , 0 . 8 , 0 . 5

respectively, and $P \left(W\right) = 0 . 8$ . Determine the probability $P \left(A_{4}\right)$ that four our more complete their papers and $P \left(A_{5}\right)$ that five or more finish.

Pfeiffer, Applied Probability, 5.3 Problems on Conditional Independence, exercise 5

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