(a) Assuming each component’s failure/success is independent of the others, what is the probability all three components fail, resulting in disaster for the flight? (b) What is the probability at least one of the components does not fail?

The probability all three fail is $p_1p_2p_3$.
The probability at least one does not fail is $1 – p_1p_2p_3$.

Explanation

• (a) By independence, the probability that all three components fail is the product of their individual failure probabilities: $P(\text{all fail})=p_1\cdot p_2\cdot p_3$.

• (b) “At least one does not fail” is the complement of “all three fail”, so

$P(\text{at least one does not fail})=1-P(\text{all fail})=1-p_1p_2p_3$.

Notes / variants

• If each component has the same failure probability $p$, then $P(\text{all fail})=p^3$ and $P(\text{at least one does not fail})=1-p^3$.
• You can also express part (b) in terms of success probabilities $s_i=1-p_i$ by expanding the complement if needed, but the compact complement form above is simplest.