The Eastmore Program is a special program to help alcoholics. In the Eastmore Pr

Question

The Eastmore Program is a special program to help alcoholics. In the Eastmore Program, an alcoholic lives at home but undergoes a two-phase treatment plan. Phase I is an intensive group-therapy program lasting 10 weeks. Phase II is a long-term counseling program lasting 1 year. Eastmore Programs are located in most major cities, and past data gave the following information based on percentages of success and failure collected over a long period of time: The probability that a client will have a relapse in phase I is 0.28; the probability that a client will have a relapse in phase II is 0.2492. However, if a client did not have a relapse in phase I, then the probability that this client will not have a relapse in phase II is 0.93. If a client did have a relapse in phase I, then the probability that this client will have a relapse in phase II is 0.71. Let A be the event that a client has a relapse in phase I and B be the event that a client has a relapse in phase II. Let C be the event that a client has no relapse in phase I and D be the event that a client has no relapse in phase II. (Enter your answers to four decimal places.)

(a) Compute P(A), P(B), P(C), and P(D).

(b) Compute P(B | A) and P(D | C).

(c) Compute P(A and B) and P(C and D).

(d) Compute P(A or B).
P(A or B) =  

(e) What is the probability that a client will go through both phase I and phase II without a relapse?


(f) What is the probability that a client will have a relapse in both phase I and phase II?


(g) What is the probability that a client will have a relapse in either phase I or phase II?

Explanation / Answer

a)

P(A)=the probability that a client will have a relapse in phase I= 0.28

P(B)=the probability that a client will have a relapse in phase II= 0.2492

P(C)=the probability that a client has no relapse in phase I=1-0.28=0.72

P(D)=the probability that a client has no relapse in phase II=1-0.2492=0.7508

(b)

P(B|A)=Probability that client did have a relapse in phase I, then the probability that this client will have a relapse in phase II= 0.71

P(D|C)=Probability that a client did not have a relapse in phase I, then the probability that this client will not have a relapse in phase I=0.93

(c) P(A and B)=P(B|A)P(A)=0.71x 0.28=0.1988

P(C and D)=P(D|C)P(C)=0.93x0.72=0.6696

(d) P(A or B) =P(A)+P(B)-P(A and B)=0.28+0.2492-0.1988=0.3304

(e) the probability that a client will go through both phase I and phase II without a relapse=

P(Ac and Bc)=P(A or B)c (From Demorgan's theorem)

=1-P(A or B)=1-0.3304=0.6696

f) the probability that a client will have a relapse in both phase I and phase II

=P(A and B)=P(B|A)P(A)=0.1988

g) probability that a client will have a relapse in either phase I or phase II=

P(A and Bc)+P(Ac and B)=P(A or B)-P(A and B)=0.3304-0.1988=0.1316

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