# Halloween candies!

Children go trick-or-treating in three mathematicians' apartments.

In MathA's apartment, a child will roll a die and the number of candies the child receives will be the same as the outcome of the die roll (1, 2, 3, 4, 5, or 6).

In MathB's apartment, a child will flip a coin 6 times and the number of candies the child receives will be the number of heads that turn up (number of heads = number of candies).

In MathC's apartment, a child will draw a single card 4 times from a deck of card. Each time a card is chosen, it will be replaced and shuffled. The child will get a candy every time when the card drawn is not a heart (candy will be given when card drawn is a diamond, club, or spade).

The distributions calculated are as follows:

MathA~Uniform (1, 6)

Number of Candies received by a child (X_A) & P(X_A)

\0 & 1/6 \1 & 1/6 \2 & 1/6 \3 & 1/6 \4 & 1/6 \5 & 1/6 \6 & 1/6

MathB~Binomial (6, 0.5)

Number of Candies received by a child (X_B) & P(X_B)

\0 & 1/64 \1 & 16/64 \2 & 15/64 \3 & 20/64 \4 & 15/64 \5 & 6/64 \6 & 1/64

MathC~Binomial (4, 0.75)

Number of Candies received by a child (X_C) & P(X_C)

\0 & 1/256 \1 & 12/256 \2 & 54/256 \3 & 108/256 \4 & 81/256

Expected number of candies collected from each mathematicians are calculated as follows:

E(X_A) = 1/2 (1+6) = 3.5

E(X_B) = 6 (0.5) = 3

E(X_C) = 4 (0.75) = 3

1. If 100 children visit the three mathematician's apartments, what is the distribution (name, mean, and variance) of the number of candies each mathematicians will hand out by the end of the night?

2. If each mathematician buys 500 pieces of candies, which of the three is most likely to run of of candies?

3. What is the expectation and variance of the total number of candies handed out by all three mathematicians? Is this distribution normally distributed? Why?