A rainstorm in Portland, Oregon, has wiped out the electricity in about 5% of the households in the city. A management team in Portland has a big meeting tomorrow, and all 6 members of the team are hard at work in their separate households, preparing their presentations. What is the probability that none of them has lost electricity in his/her household? Assume that their locations are spread out so that loss of electricity is independent among their households.

Round your response to at least 3 decimal places.

To solve this problem, we need to determine the probability that none of the 6 members of the management team has lost electricity in their households. Since the loss of electricity in each household is independent, this scenario can be modeled using the binomial distribution.

### Understanding the Problem
- The probability that a household **has** electricity (i.e., has not lost it) is \(1 - 0.05 = 0.95\).
- We need to calculate the probability that all 6 members (households) have electricity.

### Solution Using Binomial Distribution
We can model this using the binomial distribution where we want to find the probability of exactly 0 failures (loss of electricity) out of 6 trials (households).

### Binomial Distribution Formula
The probability of exactly \(k\) successes (no power outage) in \(n\) independent Bernoulli trials (households) is given by:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]

Where:
- \(n = 6\) (number of households),
- \(k = 6\) (we want all 6 to have electricity, so we consider it 6 successes),
- \(p = 0.95\) (probability that a household has electricity),
- \(\binom{n}{k} = \binom{6}{6} = 1\) (since \(\binom{n}{n} = 1\)).

### Probability Calculation
The probability that all 6 members have electricity is:

\[
P(X = 6) = 1 \times (0.95)^6
\]

Now, let's calculate this:

\[
0.95^6 \approx 0.7351
\]

### Final Answer
Thus, the probability that none of the 6 members has lost electricity is approximately **0.735**.

Would you like to explore this further or have any additional questions?

### Related Questions:
1. What is the probability that at least one member has lost electricity?
2. How would the probability change if the number of households increased to 10?
3. What if the probability of a household losing electricity increased to 10%? What is the new probability that none of them loses electricity?
4. How does the independence of events affect the probability calculation?
5. What is the expected number of households that have electricity?
6. If two team members live close to each other, how does that affect the independence assumption?
7. How would you compute the probability that exactly 2 members have lost electricity?
8. How would you approach this problem if the probability of power outage was unknown and you only had historical data?

### Tip:
For problems involving independent events, it's often easier to first calculate the probability of a single event and then raise it to the power of the number of trials (households in this case).

Learn how to calculate the probability that none of the 6 members of a management team in Portland, Oregon, have lost electricity during a rainstorm. Use the binomial distribution formula and probability theory to solve this problem.

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