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On average, Pierre, an amateur chef, drops three pieces of eggshell into every two cake batters he makes. Suppose that you buy one of his cakes.

a. The probability of finding exactly 3 pieces of eggshell in the cake is higher than the probability of finding exactly 4 pieces.

b. The probability of finding exactly 2 pieces of eggshell in the cake is higher than the probability of finding exactly 3 pieces.

c. The probability of finding exactly 4 pieces of eggshell in the cake is higher than the probability of finding exactly 5 pieces.

d. The probability of finding exactly 1 piece of eggshell in the cake is higher than the probability of finding exactly 2 pieces.

1 Answer

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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.

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