The computer that controls a bank's automatic teller machine crashes a mean of 0.3
 times per day. What is the probability that, in any seven-day week, the computer will crash less than 1
 time? Round your answer to four decimal places.



This problem involves finding the probability of a certain number of events occurring within a fixed interval of time, which is best handled using the **Poisson distribution**. The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space under the following conditions:
1. The events occur independently.
2. The average rate (mean number of events) is constant.
3. Two events cannot occur at exactly the same time.

### Given:
- Mean number of crashes per day (\(\lambda_{day}\)) = 0.3
- Number of days = 7
- Total mean number of crashes in a week (\(\lambda_{week}\)) = \( \lambda_{day} \times 7 = 0.3 \times 7 = 2.1 \)

We need to find the probability that the computer will crash less than 1 time in a 7-day week. This corresponds to finding \( P(X < 1) \) where \( X \) is a Poisson random variable with \( \lambda = 2.1 \).

### Step 1: Calculate \( P(X = 0) \)
The Poisson probability mass function is given by:
\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]
For \( k = 0 \):
\[
P(X = 0) = \frac{2.1^0 e^{-2.1}}{0!} = e^{-2.1}
\]

### Step 2: Calculate the probability
\[
P(X < 1) = P(X = 0)
\]

Let's calculate \( e^{-2.1} \). 

The probability that the computer will crash less than 1 time in a seven-day week is **0.1225**.

Would you like more details on this calculation, or do you have any further questions?

Here are five related questions you might find interesting:
1. What is the probability that the computer crashes exactly 2 times in a week?
2. How would the probability change if the mean number of crashes per day increased to 0.5?
3. What is the expected number of crashes in a 30-day month?
4. How does the Poisson distribution differ from the binomial distribution?
5. What is the probability that the computer will crash at least once in a week?

**Tip:** The Poisson distribution is often used for modeling rare events over a fixed period of time or space.

The computer that controls a bank's automatic teller machine crashes a mean of 0.3 times per day. What is the probability that, in any seven-day week, the computer will crash less than 1 time? Round your answer to four decimal places.

Explore how to calculate the probability of a bank's ATM system crashing less than 1 time in a seven-day week using the Poisson distribution. The problem involves key probability concepts and is suitable for students in Grades 10-12.

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