Final answer:
The average pieces of eggshell in a cake made by Pierre is 1.5. Probabilities for finding no eggshell pieces can be calculated using Poisson distribution, and the probability for no eggshells in six cakes is the single-cake probability raised to the power of six. While seven pieces of shell in one cake are possible, it's less probable given the average rate.
Step-by-step explanation:
Addressing the student's question regarding probabilities in the context of dropping eggshells into cake batter:
d. Average pieces of eggshell: Given that Pierre drops three pieces of eggshell into every two cake batters, the average number of pieces you'd expect to find in one cake is 1.5 (3 pieces / 2 cakes).
e. Probability of no eggshell pieces: To find this, we would assume a probability distribution such as the Poisson distribution, which is suitable for counting events (like dropping eggshells) that occur with a known constant rate independently of the time since the last event. The formula for the probability of observing k events is P(X=k) = (e-λ * λk) / k!, where λ is the average number of events. Plugging in the numbers, we find the probability for k=0.
f. Probability of no eggshell pieces over six weeks: Assuming each cake is independent of the others, we'd raise the single-cake no-shell probability to the power of six to find the probability over six weeks.
It is possible to have seven pieces of shell in a cake if the process is random and each eggshell drop is independent of the previous; however, it is less likely than having fewer pieces given the average rate provided.