Questions: Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 38% of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below. a. Find the probability that both generators fail during a power outage. (Round to four decimal places as needed.) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed. (Round to four decimal places as needed.) Is that probability high enough for the hospital? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. Yes, because both generators fail about % of the time they are needed, which is low enough to not impact the health of patients.

Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 38% of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below. a. Find the probability that both generators fail during a power outage. (Round to four decimal places as needed.) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed. (Round to four decimal places as needed.) Is that probability high enough for the hospital? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. Yes, because both generators fail about % of the time they are needed, which is low enough to not impact the health of patients.
Transcript text: submit quiz Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail $38 \%$ of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below. a. Find the probability that both generators fail during a power outage. $\square$ (Round to four decimal places as needed.) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than $1 \%$ of the time when needed. $\square$ (Round to four decimal places as needed.) Is that probability high enough for the hospital? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. Yes, because both generators fail about $\square$ $\%$ of the time they are needed, which is low enough to not impact the health of patients.
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Solution

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Solution Steps

Step 1: Probability that Both Generators Fail

To find the probability that both backup generators fail during a power outage, we use the formula for the binomial distribution:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

  • \( n = 2 \) (the number of generators),
  • \( x = 2 \) (the number of failures),
  • \( p = 0.38 \) (the probability of failure),
  • \( q = 1 - p = 0.62 \) (the probability of success).

Calculating this gives:

\[ P(X = 2) = \binom{2}{2} \cdot (0.38)^2 \cdot (0.62)^{0} = 1 \cdot 0.1444 \cdot 1 = 0.1444 \]

Thus, the probability that both generators fail is \( 0.1444 \).

Step 2: Probability of Having at Least One Working Generator

The probability of having at least one working generator is the complement of the probability that both fail:

\[ P(\text{at least one works}) = 1 - P(X = 2) = 1 - 0.1444 = 0.8556 \]

Therefore, the probability of having at least one working generator is \( 0.8556 \).

Step 3: Assessing the Probability for Hospital Needs

The hospital requires that both generators fail less than \( 1\% \) of the time. We check if the calculated probability of both failing meets this criterion:

\[ P(X = 2) = 0.1444 \]

Since \( 0.1444 \) is greater than \( 0.01 \), the probability of both generators failing is not acceptable for the hospital's needs.

Final Answer

The probability that both generators fail is \( 0.1444 \), the probability of having at least one working generator is \( 0.8556 \), and the probability of both failing is not low enough for the hospital's requirements.

\[ \boxed{0.1444, 0.8556, \text{No}} \]

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