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.