Question

During a space shot, the primary computer system is backed up by two secondary systems. They operate independently of one another and each is \(90\%\) reliable. The probability that all three systems will be operable at the time of launch is

• A. \(0.729\)
• B. \(0.81\)
• C. \(0.90\)
• D. \(0.27\)

Hint:

For independent events, \[ \boxed{ P(\text{All occur}) = \prod P_i } \] Here, \[ 0.9^3=0.729. \]

Answer 1

The Correct Option is A

Concept:

When independent events occur simultaneously, the probability that all occur is the product of their individual probabilities.

\[ P(A\cap B\cap C)=P(A)\times P(B)\times P(C) \]

Step 1: Identify the reliability of each system.

Each computer system is 90% reliable, so

\[ P(\text{system works})=0.9. \]

Step 2: Use independence.

Since the three systems operate independently,

\[ \begin{aligned} P(\text{all three work}) &=0.9\times0.9\times0.9\\ &=(0.9)^3\\ &=0.729. \end{aligned} \]

Therefore, the probability that all three systems operate successfully is

\[ P(\text{all three systems operate})=0.729. \]

Hence, the correct option is (A).