Chapter 8.6, Problem 66E

(a)

To determine

To find: the probability that during a given flight both systems function satisfactorily.

Answer to Problem 66E

The probability that both system will function satisfactorily is 0.9702.

Explanation of Solution

Given information:

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

Calculation:

Given that the probability that either system will function satisfactory for the duration of flight is $p\left(A\right)=0.985$

The probability that both system will function satisfactorily is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A\right)=\left(0.985\right)\cdot \left(0.985\right)\\ =0.9702\end{array}$

The probability that both system will function satisfactorily is 0.9702.

(a)

To determine

To find: the probability that during a given flight both systems function satisfactorily.

Answer to Problem 66E

The probability that both system will function satisfactorily is 0.9702.

Explanation of Solution

Given information:

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

Calculation:

Given that the probability that either system will function satisfactory for the duration of flight is $p\left(A\right)=0.985$

The probability that both system will function satisfactorily is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A\right)=\left(0.985\right)\cdot \left(0.985\right)\\ =0.9702\end{array}$

The probability that both system will function satisfactorily is 0.9702.

(b)

To determine

To find: the probability that during a given flight at least one system functions satisfactorily.

Answer to Problem 66E

The probability that at least one system will function satisfactorily is 0.9997.

Explanation of Solution

Given information:

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

Calculation:

Given that the probability that either system will function satisfactory for the duration of flight is $p\left(A\right)=0.985$

The probability that either system will not function satisfactorily for the duration of flight is

  $\begin{array}{c}p\left(B\right)=1-p\left(A\right)\\ =1-0.985\\ =0.9997\\ =0.015\end{array}$

The probability that at least one system will function satisfactorily is

  $\begin{array}{c}=1-\text{both}\text{failur}\\ \text{=1-}\left(0.015\right)\cdot \left(0.015\right)\\ =1-0.000225\\ =0.9997\end{array}$

The probability that at least one system will function satisfactorily is 0.9997.

(c)

To determine

To find: the probability that during a given flight both systems fail.

Answer to Problem 66E

The probability that both system fail is 0.000225.

Explanation of Solution

Given information:

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

Calculation:

Given that the probability that either system will function satisfactory for the duration of flight is $p\left(A\right)=0.985$

The probability that either system will not function satisfactorily for the duration of flight is

  $\begin{array}{c}p\left(B\right)=1-p\left(A\right)\\ =1-0.985\\ =0.9997\\ =0.015\end{array}$

The probability that both system fail is =

  $\begin{array}{c}p\left(B\right)\cdot p\left(B\right)\\ =\left(0.015\right)\cdot \left(0.015\right)\\ =0.000225\end{array}$

The probability that both system fail is 0.000225.