Each week, Stéphane needs to prepare 4 exercises for the following week's homework assignment. The number of problems he creates in a week follows a Poisson distribution with mean 6.9.
a. What is the probability that Stéphane manages to create enough exercises for the following week's homework? Round your answer to 4 decimal places.
b. Unfortunately, each week there is a 55% chance that a visiting scholar from Switzerland arrives and burdens Stéphane with research questions all week. During these weeks he only writes an average of 3.45 exercises. If Stéphane fails to write 4 exercises one week, what is the probably that he received a visiting scholar that week? Round your answer to 4 decimal places.
c. The last week of the semester, Stéphane decides to "reward" the students by no longer limiting himself to 4 exercises, and instead assigning every exercise he writes. If a student with a 60% chance of correctly answering an exercise is expected to answer 3 questions correctly, what is the probably that Stéphane did not have a visitor that week? Round your answer to 4 decimal places.
Hint: First find the number of exercises in the last week of the semester from the chance and expected value of the correct answers.
P(A and B) = P(A) * P(B)
P(A | B) = P(A and B)/p(B)
Poisson distribution equation:
P(x; μ) = (e-μ) (μx) / x!
The Attempt at a Solution
I was able to finish the first question and get the right answer but I'm having trouble on parts b and c.
For the first question:
P(x = 4) = 1 - P(x <= 3) = 1 - (P(0) + P(1) + P(2) + P(3))
Then I used the poisson distribution equation and was able to get the answer.
I think for b you need to use the conditional equation, but I'm not sure what the P(failing) would be.
I have no idea how to do c
Any help is appreciated, thanks for reading