Question 5 If you receive an average of two emails per week from your statistics professor, what is the probability that you will receive exactly one email from your professor on Monday? Round your answer to 3 decimal places. Use a point (.) instead of a comma (.) to separate whole numbers and decimals.

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UpStudy AI · Accepted Answer

Let the number of emails received follow a Poisson process. Since you receive an average of $$2$$ emails per week, the average number of emails you expect on any given day (assuming emails are equally likely on each day of the week) is

$$
\lambda_M = \frac{2}{7}.
$$

For a Poisson process, the probability of observing exactly $$k$$ events with an average rate $$\lambda$$ is given by

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

Here, we want the probability of receiving exactly one email on Monday ($$k = 1$$), so we have

$$
P(1) = \frac{\left(\frac{2}{7}\right)^1 e^{-\frac{2}{7}}}{1!} = \frac{2}{7} e^{-\frac{2}{7}}.
$$

Substituting $$\lambda_M = \frac{2}{7} \approx 0.2857$$ and computing the exponential term:

$$
e^{-\frac{2}{7}} \approx e^{-0.2857} \approx 0.751,
$$

and thus

$$
P(1) \approx 0.2857 \times 0.751 \approx 0.214.
$$

Rounded to three decimal places, the probability is

$$
\boxed{0.214}.
$$
The probability of receiving exactly one email from your professor on Monday is approximately 0.214.

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