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FairCoin_LoadedDie

Chapter 2 : Probability Concepts and Applications

1. A fair coin is tossed three times. Describe the sample space Ω.

Let X be random variable that denotes the number of heads on the first toss. Describe the probability frequency

distribution of X. Find the mean E(X) and variance Var(X) of X.

Let Y be another random variable that counts the total number of heads. Describe the probability frequency distribution

of Y , and evaluate E(Y ) and Var(Y ).

Describe the frequency distribution of the ordered pair Z = (X, Y ). Then evaluate E(Z) and Var(Z).

Homework: Describe the frequency distribution of the product W = XY . Then evaluate E(W ) and Var(W ).

2. A loaded die has the following probability distribution M :

P (M = 1) = 1
2
, P (M = 3) = P (M = 5) = P (M = 6) = 1

6
, and P (M = 2) = P (M = 4) = 0.

Notice, there is a 50% probability that the loaded die will toss the number 1.

A jar contains two fair dice and one loaded die. If one die is chosen from the jar at random, what is the probability that

the loaded die is chosen? If a chosen die is tossed twice and showed a sum of 2, find the (posterior) probability that the

die is loaded.

Apply Bayes’ formula: P (A|B) = P (B|A)P (A)
P (B)

if P (B) �= 0.

Homework: If a chosen die is tossed twice and showed a sum of 3, find the (posterior) probability that the die is loaded.

Due Sept. 3 at 5 pm

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