Chapter 4.2, Problem 36P

a.

To determine

Find the values of $P\left(A\right)$, $P\left(B\right)$, $P\left(C\right)$, and $P\left(D\right)$.

Answer to Problem 36P

The value of $P\left(A\right)$ is 0.27.

The value of $P\left(B\right)$ is 0.23.

The value of $P\left(C\right)$ is 0.73.

The value of $P\left(D\right)$ is 0.77.

Explanation of Solution

Let A be the event that a client has relapse in phase I and B be the event that a client has 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. It is noticed that event C is the complement of A and D is the complement of B.

It is given that the probability that a client will have a relapse in phase I is 0.27.  The probability that a client will have a relapse in phase II is 0.23.

That is, $P\left(A\right)=0.27$, $P\left(B\right)=0.23$.

The value of $P\left(C\right)$ and $P\left(D\right)$ is as follows:

$\begin{array}{c}P\left(C\right)=1-P\left(A\right)\\ =1-0.27\\ =0.73\end{array}$

$\begin{array}{c}P\left(D\right)=1-P\left(B\right)\\ =1-0.23\\ =0.77\end{array}$

Thus, the values of $P\left(A\right)$, $P\left(B\right)$, $P\left(C\right)$, and $P\left(D\right)$ are 0.27, 0.23, 0.73, and 0.77, respectively.

b.

To determine

Find the values of $P\left(B/A\right)$ and $P\left(D/C\right)$.

Answer to Problem 36P

The value of $P\left(B/A\right)$ is 0.70.

The value of $P\left(D/C\right)$ is 0.95.

Explanation of Solution

It is given that if a client has a relapse in phase I, then the probability of this client to have relapse in phase II is 0.70.  That is, $P\left(B/A\right)=0.70$.

Also, if the client did not have a relapse in phase I, then the probability that this client will not have a relapse in phase II is 0.95. That is, $P\left(D/C\right)=0.95$.

Thus, the values of $P\left(B/A\right)$ and $P\left(D/C\right)$ are 0.70 and 0.95, respectively.

c.

To determine

Compute $P\left(AandB\right)$ and $P\left(\text{C }and\text{ D}\right)$.

Answer to Problem 36P

The value of $P\left(AandB\right)$ is 0.189.

The value of $P\left(\text{C }and\text{ D}\right)$ is 0.657.

Explanation of Solution

The value of $P\left(AandB\right)$ is computed as follows:

$\begin{array}{c}P\left(AandB\right)=P\left(A\right)P\left(B/A\right)\\ =0.27\left(0.70\right)\\ =0.189\end{array}$

Thus, the value of $P\left(AandB\right)$ is 0.189.

The value of $P\left(\text{C }and\text{ D}\right)$ is as follows:

$\begin{array}{c}P\left(\text{C }and\text{ D}\right)=P\left(C\right)P\left(D/C\right)\\ =0.73\left(0.95\right)\\ =0.6935\end{array}$

Therefore, the value of $P\left(\text{C }and\text{ D}\right)$ is 0.6935.

d.

To determine

Find the value of $P\left(AorB\right)$.

Answer to Problem 36P

The value of $P\left(AorB\right)$ is 0.311.

Explanation of Solution

The value of $P\left(AorB\right)$ is as follows:

$\begin{array}{c}P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{ and }B\right)\\ =0.27+0.23-0.189\\ =0.311\end{array}$

Thus, the value of $P\left(AorB\right)$ is 0.311.

e.

To determine

Find the probability that a client will go through both phase I and phase II without a relapse.

Answer to Problem 36P

The probability that a client will go through both phase I and phase II without a relapse is 0.6935.

Explanation of Solution

The probability that a client will go through both phase I and phase II without a relapse can be written as $P\left(\text{C }and\text{ D}\right)$.

From Part (c), the value of $P\left(\text{C }and\text{ D}\right)$ is 0.6935.

Therefore, the probability that a client will go through both phase I and phase II without a relapse is 0.6935.

f.

To determine

Find the probability that a client will have a relapse in both phase I and phase II.

Answer to Problem 36P

The probability that a client will have a relapse in both phase I and phase II is 0.189.

Explanation of Solution

The probability that a client will have a relapse in both phase I and phase II can be written as follows:

$P\left(AandB\right)$

From Part (c), the value of $P\left(\text{C }and\text{ D}\right)$ is 0.189.

Thus, the probability that a client will have a relapse in both phase I and phase II is 0.189.

g.

To determine

Compute the probability that a client will have a relapse in either phase I or phase II.

Answer to Problem 36P

The probability that a client will have a relapse in either phase I or phase II is 0.31.

Explanation of Solution

The probability that a client will have a relapse in either phase I or phase II can be written as follows:

$P\left(AorB\right)$

From Part (d), the value of $P\left(AorB\right)$ is 0.31.

Thus, the probability that a client will have a relapse in either phase I or phase II is 0.31.