Let's approach this problem using the Poisson distribution, as it deals with the number of occurrences of an event in a fixed interval of time or space. Since the problem involves calculating the probability of a certain number of computers crashing during a severe thunderstorm, and the rate of occurrence is known, the Poisson distribution is a good fit.

Step 1: Understand the Given Information

• Rate of computer crashes: 1 computer crashes in 800 during a severe thunderstorm. This gives us a crash rate of $\lambda = \frac{1}{800}$.
• Number of computers: The company has 4,000 computers, so the expected number of crashes (mean $\lambda_{\text{total}}$) is calculated by multiplying the individual rate by the number of computers:

$\lambda_{\text{total}} = 4000 \times \frac{1}{800} = 5$

So, on average, 5 computers are expected to crash during a severe thunderstorm.

Step 2: Poisson Distribution Formula

The Poisson distribution gives the probability of exactly $k$ events occurring in a fixed interval of time or space and is given by the formula:

$P(k) = \frac{\lambda^k e^{-\lambda}}{k!}$

where:

• $\lambda$ is the mean number of occurrences (in this case, $\lambda = 5$),
• $k$ is the number of occurrences (crashed computers in this case),
• $e$ is Euler's number (approximately 2.71828).

Part a) Probability that less than 10 computers crashed

We need to find the probability that fewer than 10 computers crashed, which means we want the probability of 0, 1, 2, ..., 9 computers crashing. We compute this as the sum of the individual probabilities for $k = 0$ to $k = 9$:

$P(X < 10) = P(0) + P(1) + P(2) + \dots + P(9)$

Each of these probabilities is calculated using the Poisson formula.

Part b) Probability that exactly 10 computers crashed

For exactly 10 computers crashing, we compute the Poisson probability for $k = 10$ using the formula:

$P(X = 10) = \frac{5^{10} e^{-5}}{10!}$

Now let's compute both parts.

I'll handle the calculations for these probabilities. math

Given values

lambda_total = 5 # Mean number of crashes (5 computers) k_max = 9 # We need to sum up P(k) for k = 0 to 9 for part (a)

Part a) Compute probability that less than 10 computers crashed (sum P(k) for k = 0 to 9)

prob_less_than_10 = sum((lambda_total**k * math.exp(-lambda_total)) / math.factorial(k) for k in range(k_max + 1))

Part b) Compute probability that exactly 10 computers crashed

k_exactly_10 = 10 prob_exactly_10 = (lambda_total**k_exactly_10 * math.exp(-lambda_total)) / math.factorial(k_exactly_10)

prob_less_than_10, prob_exactly_10### Results:

a) Probability that less than 10 computers crashed:
The probability that fewer than 10 computers crashed is approximately 0.968 (or 96.8%).

b) Probability that exactly 10 computers crashed:
The probability that exactly 10 computers crashed is approximately 0.0181 (or 1.81%).

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Would you like a detailed explanation of these results or further clarification?

Here are some related questions you might find interesting:

1. How does the Poisson distribution work, and why is it used for this type of problem?
2. What would happen if the rate of computer crashes changed (for example, 1 in 500 computers)?
3. How do you interpret the meaning of the Poisson distribution in a real-world context?
4. What happens to the probabilities as the number of computers increases?
5. How would you compute the probability of more than 10 computers crashing?

Tip: The Poisson distribution is often used when events are rare and independent over a fixed interval, which is exactly the case here with computer crashes during a thunderstorm.