Question

for a parallel structure of identical components, the system can succeed if at least one of the components succeeds. assume that components fail independently of each other and that each component has a 0.11 probability of failure. complete parts (a) through (c) below.
(a) would it be unusual to observe one component fail? two components?
it ▼ be unusual to observe one component fail, since the probability that one component fails, □, is ▼ than 0.05. it ▼ be unusual to observe two components fail, since the probability that two components fail, □, is ▼ than 0.05.
(type integers or decimals. do not round.)

Explanation:

Step1: Define failure probability

Let $p = 0.11$ be the probability a single component fails.

Step2: Calculate 1-component failure prob

Probability of exactly 1 component failing (for 2 components total):
$\binom{2}{1}p(1-p) = 2 \times 0.11 \times (1-0.11) = 2 \times 0.11 \times 0.89 = 0.1958$

Step3: Compare 1-fail prob to 0.05

$0.1958 > 0.05$, so it is not unusual.

Step4: Calculate 2-component failure prob

Probability of both components failing:
$p^2 = 0.11^2 = 0.0121$

Step5: Compare 2-fail prob to 0.05

$0.0121 < 0.05$, so it is unusual.

Answer:

It would not be unusual to observe one component fail, since the probability that one component fails, $0.1958$, is greater than 0.05. It would be unusual to observe two components fail, since the probability that two components fail, $0.0121$, is less than 0.05.