Question1. 9 donuts and 3 muffins were laid out on 4 plates in such a way that there are three cakes on each plate. a) Calculate the probability that there are three muffins on one of the plates. b) Assume that it's not the case that there are three muffins on one plate. From a randomly chosen plate we move one randomly chosen cake to a different plate. Calculate the probability that after this move, the three muffins will be on the same plate. c) Calculate the expected value of the number of muffins on the first plate.

Studdy Solution
Calculate the expected value of the number of muffins on the first plate.
The expected value $E(X)$ is calculated using the formula:
$$E(X) = \sum (x \times P(x))$$
where $x$ is the number of muffins on the first plate and $P(x)$ is the probability of having $x$ muffins on the first plate.
The solutions are:
a) Probability of three muffins on one plate: $\frac{4}{\binom{12}{3}}$
b) Probability of moving a cake to create a plate with all 3 muffins: Calculated based on favorable moves.
c) Expected value of muffins on the first plate: Calculated using the expected value formula.