⧼exchistory⧽

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
  • 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].

  1. 0
  2. 1/3
  3. 1/2
  4. 2/3
  5. 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. 1/96
  2. 1/48
  3. 3/96
  4. 1/24
  5. 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].

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

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

  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]
  • 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?

  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]
  • 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]?

  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]
  • 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?

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

  1. 0.04
  2. 0.05
  3. 0.06
  4. 0.07875
  5. 0.11
  • Created by Admin, May 31'22