5.2 Problems on conditional independence

Suppose $\{A,B\}\mathrm{ci}|C$ and $\{A,B\}\mathrm{ci}|C^c$ , $P\left(C\right)=0.7$ , and

Show whether or not the pair $\{A,B\}$ is independent.

Suppose $\{A_1,A_2,A_3\}\mathrm{ci}|C$ and $\mathrm{ci}|C^c$ , with $P\left(C\right)=0.4$ , and

Determine the posterior odds $P\left(C\right|A_1{A}_2^c{A}_3)/P\left(C^c\right|A_1{A}_2^c{A}_3)$ .

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 latecold 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 ofbreaking the record are:

• Good weather (no front): 0.75, 0.80, 0.65, 0.70, 0.85
• 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 B ₃ is the event of three or more, $P\left(B_3\right)=P\left(B_3\right|W)P\left(W\right)+P\left(B_3\right|W^c)P\left(W^c\right)$ .

A device has five sensors connected to an alarm system. The alarm is given if three or more of the sensors trigger a switch. If a dangerous condition ispresent, each of the switches has high (but not unit) probability of activating; if the dangerous condition does not exist, each of the switches has low (but notzero) probability of activating (falsely). Suppose $D=$ the event of the dangerous condition and $A=$ the event the alarm is activated. Proper operation consists of $AD\bigvee A^c{D}^c$ . Suppose $E_i=$ the event the i th unit is activated. Since the switches operate independently, we suppose

Assume the conditional probabilities of the E ₁ , given D , are 0.91, 0.93, 0.96, 0.87, 0.97, and given D ^c , are 0.03, 0.02, 0.07, 0.04, 0.01, respectively. If $P\left(D\right)=0.02$ , what is the probability the alarm system acts properly? Suggestion . Use the conditional independence and the procedure ckn.

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 i th 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 $\{E_i:1\le i\le 7\}\mathrm{ci}|W$ and $\mathrm{ci}|W^c$ . Suppose

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.