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].
- 0
- 1/8
- 1/4
- 1/2
- 3/4
- Created by Admin, May 31'22
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].
- 0
- 1/3
- 1/2
- 2/3
- 1
- Created by Admin, May 31'22
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.
- 1/96
- 1/48
- 3/96
- 1/24
- 1/12
- Created by Admin, May 31'22
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].
- 3/8
- 1/2
- 5/8
- 3/4
- 1
- Created by Admin, May 31'22
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].
- 0
- 1/4
- 1/2
- 3/4
- 1
- Created by Admin, May 31'22
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?
- [math]\operatorname{P}(B) = 1 [/math]
- [math]0 \lt \operatorname{P}(B) \lt 1/4 [/math]
- [math]1/4 \leq \operatorname{P}(B) \leq 1/2 [/math]
- [math]\operatorname{P}(B) = 0 [/math]
- [math]1/2 \lt \operatorname{P}(B) \lt 1[/math]
- Created by Admin, May 31'22
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?
- [math]\operatorname{P}(\cap_{n=1}^{\infty} A_n) = 0 [/math]
- [math]0 \lt \lim_{n\rightarrow \infty} \operatorname{P}(A_n) \lt 1/3 [/math]
- [math]\lim_{n\rightarrow \infty} \operatorname{P}(A_n) = \operatorname{P}(\cap_{n=1}^{\infty} A_n)[/math]
- [math]\lim_{n\rightarrow \infty} \operatorname{P}(A_n)[/math] cannot be determined
- [math]\lim_{n\rightarrow \infty} \operatorname{P}(A_n) \gt \operatorname{P}(\cap_{n=1}^{\infty} A_n) [/math]
- Created by Admin, May 31'22
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]?
- [math]\cup_{n=1}^{\infty}A_n[/math]
- [math]\cup_{N=1}^{\infty}\cap_{n=1}^N A_n[/math]
- [math]\cup_{N=1}^{\infty} \cap_{n=N}^{\infty}A_n[/math]
- [math]\cup_{N=1}^{\infty} \cap_{n=N}^{\infty}A_n^c[/math]
- [math]\cup_{N=1}^{\infty} \cap_{n=1}^{N}A_n^c[/math]
- Created by Admin, May 31'22
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?
- 0.4
- 0.5
- 0.6
- 0.8
- 1
- Created by Admin, May 31'22
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.
- 0.04
- 0.05
- 0.06
- 0.07875
- 0.11
- Created by Admin, May 31'22