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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?

Unsure answers:

  1. MathA: E(Y_A) = (7/2) * 100 = 350

MathB: E(Y_B) = 3 * 100 = 300

MathC: E(Y_C) = 3 * 100 = 300

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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many would consider your post rude because it simply demands an answer, instead of requesting help, so please consider rewriting. –  Zev Chonoles Feb 16 '13 at 22:12
    
i hope for the child it rolls a dice not a die –  Dominic Michaelis Feb 16 '13 at 22:14
3  
@DominicMichaelis Nope, "die" is the singular form. One die, two dice. –  Arthur Feb 16 '13 at 23:20
2  
What are the parents doing while all this is going on? –  Will Jagy Feb 17 '13 at 0:13
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@DominicMichaelis, the two main uses of "die" as a noun are as a cube for games or as an engraved device for stamping coins or medals. Meanwhile, "dye" as either verb or noun refers to coloring cloth. –  Will Jagy Feb 17 '13 at 0:26

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