Exam P Practice Problem 98 – flipping coins

Problem 98-A

Coin 1 is an unbiased coin, i.e. when flipping the coin, the probability of getting a head is 0.5. Coin 2 is a biased coin such that when flipping the coin, the probability of getting a head is 0.6. One of the coins is chosen at random. Then the chosen coin is tossed repeatedly until a head is obtained.

Suppose that the first head is observed in the fifth toss. Determine the probability that the chosen coin is Coin 2.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.2856$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.3060$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.3295$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.3564$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.3690$

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Problem 98-B

Box 1 contains 3 red balls and 1 white ball while Box 2 contains 2 red balls and 2 white balls. The two boxes are identical in appearance. One of the boxes is chosen at random. A ball is sampled from the chosen box with replacement until a white ball is obtained.

Determine the probability that the chosen box is Box 1 if the first white ball is observed on the 6th draw.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.7530$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.7632$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.7825$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.7863$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.7915$

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probability exam P

actuarial exam

math

Daniel Ma

mathematics

geometric distribution

Bayes

$\copyright$ 2017 – Dan Ma

Exam P Practice Problem 94 – Tracking High School Students

Problem 94-A

A researcher tracked a group of 900 high school students taking standardized tests in math and chemistry. Some of the students were given after-school tutoring before the tests (in both subjects) and the rest of the students received no tutoring. The following information is known about the test results:

• 510 of the students passed math test and 475 of the students passed chemistry test.
• Of the students who failed both subjects, there were 20% more students who did not receive tutoring than there were students who received tutoring.
• Of the students who failed chemistry and had tutoring, there were 99 more students who failed math than there were students who passed math.
• Of the students who failed chemistry and had no tutoring, there were 4 more students who failed math than there were students who passed math.
• There were 126 students who failed math and passed chemistry.

Determine the probability that a randomly selected student from this group had tutoring given that the student passed both subjects.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.6810$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.6828$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.6859$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.6877$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.6989$

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Problem 94-B

An insurance company tracked a group of 800 insureds for 2 years. It was found that 560 of the insureds had no claims in year 1 and 380 of the insureds had no claims in year 2. Of the insureds who had no claims in both years, there were four times as many male insureds than there were female insureds. Furthermore, there were 230 male insureds who had no claims in year 2 and there were 53 females insureds who had claims in both years. It is also known that there were 85 male insureds who had claims in year 1.

Determine the number of insureds who had no claims in year 1 but had claims in year 2.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 320$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 347$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 369$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 420$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 560$

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$\copyright \ 2016 \ \ \text{ Dan Ma}$

Exam P Practice Problem 91 – Reviewing a Group of Policyholders

Problem 91-A

A life insurance actuary reviewed a group of policyholders whose policies or contracts were inforce as of last year. The actuary found that 12% of the policyholders who had only a life insurance policy did not survive to this year and that 7.5% of the policyholders who had only an annuity contract did not survive to this year. The actuary also found that 5.9% of the policyholders who had both a life insurance policy and an annuity contract did not survive to this year.

In this group of policyholders, 65% of the policyholders had a life insurance policy and 57% of the policyholders had an annuity contract. Furthermore, each policyholder in this group either had a life insurance policy or an annuity contract.

What is the percentage of the policyholders that did not survive to this year?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 7.8 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 9.0 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 12.0 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 13.4 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 25.4 \%$

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Problem 91-B

A sport coach in a university tracks a group of athletes. The coach finds that 36% of the athletes who play soccer only are first year university students and that 20% of the athletes who are involved only in track and field are first year university students. The coach also discovers that 27% of the athletes participates in both soccer and track and field are first year university students.

According to university records, 45% of the athletes in this group play soccer and 68% of the athletes in this group participate in track and field. Each of the athletes in this group either plays soccer or participates in track and field.

Out of this group of athletes, what is the percentage of the athletes that are not first year university students?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 64 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 67 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 70 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 74 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 80 \%$

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$\copyright \ 2014 \ \ \text{ Dan Ma}$

Exam P Practice Problem 85 – Supplemental Coverages for Employees

Problem 85-A

A large employer offers three supplemental benefits in addition to the major medical benefit – dental, vision and group life insurance – to its employees. The following information is known.

• Thirty five percent of the employees choose dental and vision.
• Twelve percent of the employees choose all three supplemental benefits.
• Fifty six percent of the employees choose exactly two of the three supplemental benefits.
• Sixteen percent of the employees choose neither vision nor dental.
• Five percent of the employees choose none of the three supplemental benefits.

One employee is chosen at random. What is the probability that this employee has group life insurance benefit?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.32$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.56$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.63$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.65$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.68$

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Problem 85-B

A professor tracks a group of students taking astrophysics, biostatistics and chemistry. The following information is known.

• Forty five percent of the students pass astrophysics and biostatistics.
• Nineteen percent of the students pass biostatistics and chemistry only.
• Twenty two percent of the students pass neither biostatistics nor chemistry.
• Sixty seven percent of the students pass biostatistics.
• Thirty four percent of the students pass astrophysics and biostatistics only.

One student is chosen at random. What is the probability that this student passes chemistry?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.35$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.40$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.41$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.47$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.53$

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$\copyright \ 2014 \ \ \text{ Dan Ma}$

Exam P Practice Problem 81 – A Medical Survey

Problem 81-A

A medical researcher tracked a group of 10,000 patients with heart disease and diabetes. The researcher found that sixty five percent of these patients have health insurance coverage. She also found that thirty percent and twenty percent of the patients with health insurance coverage have heart disease and diabetes, respectively. Of the patients with health insurance coverage, ten percent have both heart disease and diabetes. Furthermore, the medical researcher found that forty percent and twenty five percent of the patients with no health insurance coverage have heart disease and diabetes, respectively. Twenty percent of the patients with no health insurance coverage have both heart disease and diabetes.

What is the number of patients that have neither heart disease nor diabetes?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2600$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 4175$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 5825$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 8500$

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Problem 81-B

A large high school offers three sporting activities – swimming, running and basketball. Thirty percent of the students participate in swimming. Of the students participating in swimming, fifty percent participate in running, thirty percent participate in basketball and five percent participate in both running and basketball. Of the students not participating in swimming, twenty percent participate in running, forty percent participate in basketball and ten percent participate in both running and basketball.

What is the percentage of the students that participate in exactly two of the sports?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 10 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 22.5 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 28 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 35 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 57.5 \%$

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$\copyright \ 2014$

Exam P Practice Problem 78 – Tracking A Group of Insureds

Problem 78-A

An insurance company tracked a group of 625 insureds for 2 years. It was found that 370 of the insureds had no claims in year 1 and 395 of the insureds had no claims in year 2. Furthermore, there were five times as many insureds who had no claims in both years than there were insureds who had claims in both years.

Select an insured at random from this group. What is the probability that the randomly selected insured had no claims in both years?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.23$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.44$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.47$

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Problem 78-B

An insurance company tracked a group of 1000 insureds for 2 years. It was found that 550 of the insureds had no claims in year 1 and 680 of the insureds had at least one claim in year 2. Of the insureds who had at least one claim in year 1, 306 insureds had at least one claim in year 2.

Select an insured at random from this group. Given that this insured had no claims in year 1, what is the probability that the selected insured had no claims in year 2?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.176$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.230$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.320$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.418$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.764$

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$\copyright \ 2013 \ \ \text{Dan Ma}$