Chapter 8.6, Problem 62E

(a).

To determine

To find:

The probability that the both systems functions satisfactorily.

Answer to Problem 62E

The required probability is $0.970225$ .

Explanation of Solution

Given:

A space vehicle has an independent backup system for one of its communications networks. The probability that either system will function satisfactory for the duration of a flight is $0.985$ .

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability that either system will function satisfactory for the duration of a flight is $0.985$ .

So, the probability of both systems functioning satisfactory is:

  $\begin{array}{c}P=0.985\cdot 0.985\\ =0.970225\end{array}$

Conclusion:

Hence, the required probability is $0.970225$ .

(b).

To determine

To find:

The probability that the at least one system functions satisfactorily.

Answer to Problem 62E

The required probability is $0.999775$ .

Explanation of Solution

Given:

A space vehicle has an independent backup system for one of its communications networks. The probability that either system will function satisfactory for the duration of a flight is $0.985$ .

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability that either system will function satisfactory for the duration of a flight is $0.985$ .

The probability of one of the systems is falling is:

  $\begin{array}{c}P\left(falling\right)=1-0.985\\ =0.015\end{array}$

So, the probability of both systems falling is:

  $\begin{array}{c}P=0.015\cdot 0.015\\ =0.000225\end{array}$

Therefore, the probability of at least one system is functioning satisfactorily is $1-0.000225=0.999775$ .

Conclusion:

Hence, the required probability is $0.999775$ .

(c).

To determine

To find:

The probability that the both systems fall.

Answer to Problem 62E

The required probability is $0.000225$ .

Explanation of Solution

Given:

A space vehicle has an independent backup system for one of its communications networks. The probability that either system will function satisfactory for the duration of a flight is $0.985$ .

Concept Used:

Apply the formula of the probability:

  $\begin{array}{c}P\left(A\right)=\frac{NumberofFavourableOutcome}{TotalNumberofFavourableOutcome}\\ =\frac{n\left(A\right)}{n\left(S\right)}\end{array}$

Calculation:

The probability that either system will function satisfactory for the duration of a flight is $0.985$ .

The probability of one of the systems is falling is:

  $\begin{array}{c}P\left(falling\right)=1-0.985\\ =0.015\end{array}$

So, the probability of both systems falling is:

  $\begin{array}{c}P=0.015\cdot 0.015\\ =0.000225\end{array}$

Conclusion:

Hence, the required probability is $0.000225$ .