Final answer:
To find the probabilities, use the Poisson distribution formula. For exactly three official engagements, use the formula with the average rate of engagements. For more than seven engagements, subtract the probability of having at most seven. For at most two engagements in both weeks, square the probability of at most two engagements in one week.
Step-by-step explanation:
To find the probability that the Duchess will have exactly three official engagements, we need to use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time, given the average rate of occurrence.
Let's denote the average rate of official engagements per week as lambda (λ), which is given as 4.75. We can plug this value into the Poisson distribution formula:
P(X = k) = (e-λ * λk) / k!
Now, we can calculate the probability of having exactly three official engagements:
P(X = 3) = (e-4.75 * 4.753) / 3!
To find the probability that the Duchess will have more than seven official engagements, we need to calculate the sum of probabilities from X = 8 to infinity:
P(X > 7) = 1 - P(X <= 7)
Lastly, to find the probability that the Duchess will have at most two official engagements in both weeks, we can square the probability of having at most two official engagements in one week:
P(X <= 2 in both weeks) = P(X <= 2 in one week) ^ 2