For problems 1-10 consider the experiment where two dice are thrown simultaneously. If both die do not have a clear upward facing side, the dice are thrown again. Each dice has six faces, numbered 1-6. Assume that the dice are fair and that each side is equally likely to occur and the dice do not influence each other.

- What is the probability that the sum of the upward facing sides of both dice is equal to two?
- What is the probability the sum of the upward facing sides of both dice is equal to three?
- What is the probability the sum of the upward facing sides of both dice is equal to seven?
- What is the probability the upward facing sides of the two die are the same?
- What is the probability the upward facing sides of the two die are different?
- What is the probability that at least one of the upward facing sides of the dice is a four, five or six?
- What is the probability that exactly one of the upward facing sides of the dice is a four, five or six?
- What is the probability that none of the upward facing sides of the dice is a four, five or six?
- After throwing the two die, you observe that one of them is a six. What is the probability that the second dice is also a six?
- After throwing the two die, you observe that one of them is a six. What is the probability that the second dice is not a six?

Given the following probabilities, answer questions 11-25. Remember that Ā is called the complement of A, and P(Ā) = 1- P(A).

- What is the P(A)?
- What is the P(B)?
- What is the P(Ā)?
- What is the P(A and B)?
- What is the P(A and )?
- What is the P(A and Ā)?
- What is the P(A|B)?
- What is the P(Ā|B)?
- What is the P(B|A)?
- What is P(A| Ā)?
- What is the P(A or B)?
- What is the P(A or Ā)?
- What is the P(A and Ā)?
- Are A and Ā independent of each other?
- Are A and B independent of each other?

For problems 26-30, suppose P(A) = 0.54 and P(B|A) = 0.27 and P(B| Ā) = 0.68

- What is P(Ā)?
- What is P(A| B)?
- What is P(A|)?
- What is P(B)? =
- Is A independent of B?

For problems 31-35, suppose P(A) = 0.42 and P(B|A) = 0.34 and P(B| Ā) = 0.34 .

- What is P(Ā)?
- What is P(A| B)?
- What is P(A|)?
- What is P(B)?
- Is A independent of B?

**Note:** For the following two Bonus Questions, please manually add any extra credit points to the student’s grade.

- (Bonus Question) Suppose X is the amount of money a person earns in an hour where P(X = $5) =0.2, P(X =$10) = 0.1, P(X=$15) =0.5 and P(X=$20) = 0.2. How much money would the person earn on average?
- (Bonus Question) If the payoff of rolling two dice is the sum of the upper face of the two dice, what is the expected payoff of one roll of the two dice?

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