A boiler has five identical relief valves. The probability that any particular valve will open on demand is 0.92. Assume indepen

Question

Vadim26

3 months ago

7

A boiler has five identical relief valves. The probability that any particular valve will open on demand is 0.92. Assume indepen

dent operation of the valves. Calculate P(at least one valve opens). (Round your answer to eight decimal places.)

Mathematics

1 answer:

Svet_ta [12.7K] · Answer 1 · 2026-02-22T11:05:33+03:00

Answer:

P(at least one valve triggers) = 0.67232.

Step-by-step explanation:

Consider a boiler with five identical relief valves, where the probability of each valve opening on request is 0.92, assuming they operate independently.

This scenario models a Binomial distribution;

$P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r}; x = 0,1,2,3,.....$

where n = total trials (number of samples) = 5 identical relief valves

r = success count = at least one valve opens

p = success probability, which in this case is 0.92, the chance of a specific valve opening when needed.

Let X = count of valves that open on demand

This indicates X ~ $Binom(n=5, p=0.92)$

The probability that at least one valve opens can be expressed as = P(X $\geq$ 1)

P(X $\geq$ 1) = 1 - Probability that none of the valves open

= 1 - P(X = 0)

= 1 - $\binom{5}{0}0.92^{0} (1-0.92)^{5-0}$

= $1-(1 \times 1 \times 0.08^{5})$

= 1 - 0.32768 results in 0.67232

Consequently, the chance that at least one valve activates is 0.67232.