Question

Assume that the engine component of a spacecraft consists of two engines in parallel. If the main engine is 95% reliable, the backup is 80% reliable, and the engine component as a whole is 99% reliable.
a) Find the probability that the main engine will fail but the backup will be operable.
b) Find the probability that the backup engine will fail but the main engine will be operable.
c) What is the probability that the engine component will fail?
d) What is the probability that in an engine system such as that described the backup engine will function given that the main engine fails?

Answer 1

Answer:

a) 0.04

b) 0.19

c) 0.01

d) 0.8

Step-by-step explanation:

Hi,

Let's make our data, considering two rules of probability:

1. The total probability of any event is 1.
2. Percentages are converted to decimal.

• P(M) = 0.95
• P(M')=0.05
• P(B)=0.80
• P(B')=0.20

- where M → Main Engine works, B → Back Up Engine works, M' → Main Engine will not work and B' → Back Up Engine will not work.

Remember for cases with and, we multiply; for cases with or, we use addition.

a) For this part, we know that Main Engine will work and Back-up engine will not work:

P(M' and B): 0.05 x 0.80 = 0.04

b) Back up will not work and Main will work:

P(M and B') = 0.20 x 0.95 = 0.19

c) We know the probability of entire component working, which is 0.99.

So to find the probability of entire component failing, we need to subtract 0.99 from 1. (Since the total probability is always 1)

P(Entire component will fail) = 1 - 0.99 = 0.01

d) This is a typical case of conditional probability, to calculate a conditional probability we use the following formula:  $P(A|B) = \frac{P(A\ and\ B)}{P(B)}$

Where, $P(A|B)$ means the probability of A, given B.

Simply using this, we calculate the $P(B|M')$:

$P( B\ and\ M') = 0.04$  (as calculated in part a)

$P(M')=0.05$ (as calculated for the data)

Modifying the formula to our needs: $P(B|M') = \frac{P(B\ and\ M')}{P(M')}$

⇒$P(B|M') = \frac{0.04}{0.05} = 0.80$ Ans.

I hope this answers all your queries.