Suppose the following is true for three events [math]A,B,C[/math]:

• [math]2\operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B) [/math]
• [math]\operatorname{P}(A) = 2\operatorname{P}(B) [/math]
• [math]\operatorname{P}(B) = 2\operatorname{P}(C) [/math]
• [math]\operatorname{P}(A \cup B \cup C) = 1 [/math]
• [math]\operatorname{P}((A \cup B) \cap C) = 0 [/math]

Determine [math]\operatorname{P}(A-B)[/math].

1. 0
2. 1/8
3. 1/4
4. 1/2
5. 3/4

Suppose the following holds:

• [math]\operatorname{P}(A-B) = \operatorname{P}(B-A) [/math]
• [math]\operatorname{P}(A \cup B) = 1 [/math]

Determine [math]\operatorname{P}(A)[/math].

1. 0
2. 1/3
3. 1/2
4. 2/3
5. 1

The probability that [math]n \geq 1[/math] claims are generated for an insurance policy equals [math]2^{-n}[/math]. Determine the probability that the number of claims generated for the policy is odd and greater than 6.

1. 1/96
2. 1/48
3. 3/96
4. 1/24
5. 1/12

Suppose the following is true:

• [math]\operatorname{P}(A \cup B) = \operatorname{P}(A) + \operatorname{P}(B) [/math]
• [math]\operatorname{P}(A \cap C ) = \operatorname{P}(A)\operatorname{P}(C) [/math]
• [math]\operatorname{P}(B \cap C) = \operatorname{P}(B)\operatorname{P}(C) [/math]
• [math]\operatorname{P}(A) = \operatorname{P}(B) = \operatorname{P}(C) = 1/4 [/math]

Determine [math]\operatorname{P}(A \cup B \cup C) [/math].

1. 3/8
2. 1/2
3. 5/8
4. 3/4
5. 1

Suppose the following is true:

• [math]\operatorname{P}(A \cap B) = \operatorname{P}(A)\operatorname{P}(B) [/math]
• [math]\operatorname{P}(A) = 1/4 [/math]

Determine [math]\operatorname{P}(A^c | B)[/math].

1. 0
2. 1/4
3. 1/2
4. 3/4
5. 1

Suppose that [math]A_n[/math] denotes the event that insurance policy [math]n[/math] has a claim and [math]\operatorname{P}(A_n) = 1-3^{-n} [/math]. If [math]B[/math] denotes the event that one of the policies has no claim, which of the following statements is true?

1. [math]\operatorname{P}(B) = 1 [/math]
2. [math]0 \lt \operatorname{P}(B) \lt 1/4 [/math]
3. [math]1/4 \leq \operatorname{P}(B) \leq 1/2 [/math]
4. [math]\operatorname{P}(B) = 0 [/math]
5. [math]1/2 \lt \operatorname{P}(B) \lt 1[/math]

Suppose that we have a sequence of events [math]A_1,A_2,\ldots [/math] with [math]A_{n+1} \subset A_{n} [/math] for all [math] n [/math]. If [math]\operatorname{P}(A_n) \geq 1/3 [/math] for all [math]n [/math], which of the following statements is true?

1. [math]\operatorname{P}(\cap_{n=1}^{\infty} A_n) = 0 [/math]
2. [math]0 \lt \lim_{n\rightarrow \infty} \operatorname{P}(A_n) \lt 1/3 [/math]
3. [math]\lim_{n\rightarrow \infty} \operatorname{P}(A_n) = \operatorname{P}(\cap_{n=1}^{\infty} A_n)[/math]
4. [math]\lim_{n\rightarrow \infty} \operatorname{P}(A_n)[/math] cannot be determined
5. [math]\lim_{n\rightarrow \infty} \operatorname{P}(A_n) \gt \operatorname{P}(\cap_{n=1}^{\infty} A_n) [/math]

Suppose we have an infinite sequence of events [math]A_1, A_2, \ldots [/math]. If the event [math]A[/math] is the event that only finitely many of the events occurred, which of the following expressions represents [math]A[/math]?

1. [math]\cup_{n=1}^{\infty}A_n[/math]
2. [math]\cup_{N=1}^{\infty}\cap_{n=1}^N A_n[/math]
3. [math]\cup_{N=1}^{\infty} \cap_{n=N}^{\infty}A_n[/math]
4. [math]\cup_{N=1}^{\infty} \cap_{n=N}^{\infty}A_n^c[/math]
5. [math]\cup_{N=1}^{\infty} \cap_{n=1}^{N}A_n^c[/math]

The lifetime of a battery brand has the following properties:

• The probability that the battery will die in less than a year is 0.2
• The probability that the battery will die in less than two years is 0.5
• The probability that the battery will die in less than three years is 0.9

Now suppose that you bought a new battery exactly two years ago, what is the probability that it will die in the coming year?

1. 0.4
2. 0.5
3. 0.6
4. 0.8
5. 1

A sports network showcasing the winter olympics has determined the following about its viewership:

• 45% watched hockey
• 35% watched ski
• 50% watched figure skating
• 11% watched figure skating and hockey
• 16% watched hockey and ski
• 25% watched figure skating and ski
• 5% didn't watch any of these sports

Determine the percentage of viewers that watched all three events.

1. 0.04
2. 0.05
3. 0.06
4. 0.07875
5. 0.11