Exam P Practice Test

95 Questions | By Aaronsmith0924 | Last updated: Mar 21, 2022 | Total Attempts: 288

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Taken from the four 1/P Exams from The Infinite Actuary.

Questions and Answers

1.

75% of the customers of ACME Mutual Insurance have auto insurance, and 40% have homeowners insurance. What is the maximum possible probability that a randomly selected customer with auto insurance does not have homewoners insurance?
- A.
  
  20%
- B.
  
  40%
- C.
  
  60%
- D.
  
  80%
- E.
  
  100%
2.

Suppose is a normal random variable with and coefficient of variation 3. Find .
- A.
  
  0.05
- B.
  
  0.34
- C.
  
  0.57
- D.
  
  0.66
- E.
  
  0.95
3.

The probability that Rafael Nadal wins a tennis match in straight sets is 70%. Assuming that the outcome of each match is independent, what is the probability that in his next 7 matches he will win in straight sets at least 5 times?
- A.
  
  0.13
- B.
  
  0.33
- C.
  
  0.44
- D.
  
  0.65
- E.
  
  0.96
4.

and have joint density given by Find .
- A.
  
  0.7
- B.
  
  0.9
- C.
  
  1.2
- D.
  
  1.4
- E.
  
  1.6
5.

The joint cdf of is given by . for . Find .
- A.
  
  0.05
- B.
  
  0.17
- C.
  
  0.19
- D.
  
  0.41
- E.
  
  0.61
6.

Let be the number of rolls of a fair die before getting a 6 and let be the number of rolls before the first even number. Find .
- A.
  
  5
- B.
  
  6
- C.
  
  7
- D.
  
  8
- E.
  
  9
7.

Suppose and are bivariate random variables with and . If find .
- A.
  
  0.40
- B.
  
  0.45
- C.
  
  0.53
- D.
  
  0.55
- E.
  
  0.60
8.

Suppose that I roll two independent dice one red and one blue. Let be the event that the blue die is even the event that the red die is even and the event that the sum is even. Which of the following is true?
- A.
  
  None of them are independent.
- B.
  
  A and B are pairwise independent, but neither is pairwise independent of C.
- C.
  
  A and C are pairwise independent, as are B and C, but A and B are not pairwise independent.
- D.
  
  All three pairs are pairwise independent, but it is not true that all three are mutually independent.
- E.
  
  All three are mutually independent.
9.

For the joint density of is given by . Find .
- A.
  
  0.75
- B.
  
  0.78
- C.
  
  0.81
- D.
  
  0.84
- E.
  
  0.87
10.

A student who is taking a 30-question multiple choice test knows the answer to 24 of the questions. Whenever the student doesn't know the answer to a question, he chooses uniformly from one of the 5 choices. Given that the student gets a randomly chosen question right, what is the probability that the student guessed on the question?
- A.
- B.
- C.
- D.
- E.
11.

If and are i.i.d. exponential random variables with mean 3 what is ?
- A.
  
  27
- B.
  
  54
- C.
  
  81
- D.
  
  108
- E.
  
  135
12.

Suppose that and are independent Poisson random variables with and . Find .
- A.
  
  0.17
- B.
  
  0.21
- C.
  
  0.25
- D.
  
  0.29
- E.
  
  0.34
13.

Let be a randomly chosen integer with . What is the probability that is not divisible by 7 11 or 13?
- A.
  
  0.66
- B.
  
  0.69
- C.
  
  0.72
- D.
  
  0.75
- E.
  
  0.78
14.

Insurance losses in a given year have a lognormal distribution with where is a normal random variable with mean 3.9 and standard deviation 0.8. If a $100 deductible and a $50 benefit limit are imposed what is the probability that the insurance company will pay the benefit limit given that a loss exceeds the deductible?
- A.
  
  0.10
- B.
  
  0.27
- C.
  
  0.43
- D.
  
  0.66
- E.
  
  0.88
15.

A fair 6-sided die is rolled 1,000 times. Using a normal approximation with a continuity correction, what is the probability that the number of 3's that are rolled is greater than 150 and less than 180?
- A.
  
  0.78
- B.
  
  0.81
- C.
  
  0.84
- D.
  
  0.88
- E.
  
  0.95
16.

Four red dice and six blue dice are rolled. Assuming that all ten dice are fair six-sided dice, and rolls are independent, what is the probability that exactly three of the red dice are even, and exactly two of the blue dice come up ones?
- A.
  
  0.05
- B.
  
  0.10
- C.
  
  0.16
- D.
  
  0.21
- E.
  
  0.27
17.

A life insurance company classifies its customers as being either high or low risk. If 20% of the customers are high risk, and high risk customers are three times as likely as low risk customers to file a claim, what percentage of claims that are filed come from high risk customers?
- A.
  
  30%
- B.
  
  37%
- C.
  
  43%
- D.
  
  54%
- E.
  
  60%
18.

Suppose that are random variables with and . If for what is where .
- A.
  
  0
- B.
  
  100
- C.
  
  1,000
- D.
  
  5,050
- E.
  
  10,000
19.

The moment generating function of is . Find .
- A.
  
  1
- B.
  
  2
- C.
  
  3
- D.
  
  4
- E.
  
  5
20.

The cdf of a random variable satisfies for . Find .
- A.
  
  0.22
- B.
  
  0.36
- C.
  
  0.51
- D.
  
  0.64
- E.
  
  0.78
21.

The density of is proportional to for and is 0 otherwise. Find the 80th percentile of .
- A.
  
  0.9
- B.
  
  1.3
- C.
  
  1.8
- D.
  
  2.3
- E.
  
  2.8
22.

If is a Poisson random variable with then what is the probability that will be within 1 standard deviation of ?
- A.
  
  0.08
- B.
  
  0.29
- C.
  
  0.47
- D.
  
  0.63
- E.
  
  0.81
23.

The joint density of and is for and . Find .
- A.
  
  1.1
- B.
  
  1.3
- C.
  
  1.5
- D.
  
  1.7
- E.
  
  1.9
24.

Suppose that are i.i.d. uniform random variables on the interval . Let denote the average of through and let and denote the standard deviation and mean of . Find the probability that the minimum and maximum of both differ from by less than .
- A.
  
  0.00001
- B.
  
  0.00032
- C.
  
  0.00057
- D.
  
  0.00083
- E.
  
  0.00115
25.

Find the mode of a Poisson random variable with mean 1.5.
- A.
  
  0
- B.
  
  0.5
- C.
  
  1
- D.
  
  1.5
- E.
  
  2