A police department has deployed a fleet of 5 police officers to give speeding tickets during the last day of the current month. The police department wants to give a fixed fine for every speeding infraction. Each police officer has to measure the speed of 70 vehicles and the probability that a vehicle goes above the speed limit is 0.15.

Using the normal approximation and the continuity correction, determine the smallest fine, rounded to the nearest integer, that ensures the department will raise at least $7,000 with 95% certainty.

• $171
• $180
• $185
• $190
• $200

An insurer has classified a pool of policies into two classes: class A and class B. The probability of observing more than one claim during a single coverage period is zero for all policies and claim size is constant for all policies:

Using the normal approximation for aggregate losses, the insurer sets insurance rates at the lowest level that guarantees a profit 95% of the time.

Determine the rate for class B policyholders assuming that the expected profit % is the same for all policies.

• $12.16
• $12.77
• $13.31
• $26.09
• $30.65

A portfolio of annual policies are classified into two classes: class A and class B. You are given the following assumptions about the policies:

• Claim size is always 100 for class A policies and 200 for class B policies
• The probability of observing more than one claim per policy is zero for all policies in the portfolio
• The probability of observing a single claim in a year is 0.05 for class A policies and 0.03 for class B policies
• 25% of the policies in the portfolio belong to class A
• The expected loss for class A policies is 0.9 times the premium and the expected loss for class B policies is 0.8 times the premium.

Using the normal approximation, determine the smallest number of policies required in the portfolio that is divisible by 4 and large enough that the portfolio is profitable at least 95% of the time.

1. 1,020
2. 1,032
3. 1,672
4. 1,680
5. 1,688

A student takes a multiple-choice test with 40 questions. The probability that the student answers a given question correctly is 0.5, independent of all other questions. The probability that the student answers more than [math]N[/math] questions correctly is greater than 0.10. The probability that the student answers more than [math]N+1[/math] questions correctly is less than 0.10.

Calculate [math]N[/math] using a normal approximation with the continuity correction.

• 23
• 25
• 32
• 33
• 35

A company provides a death benefit of 50,000 for each of its 1000 employees. There is a 1.4% chance that any one employee will die next year, independent of all other employees. The company establishes a fund such that the probability is at least 0.99 that the fund will cover next year’s death benefits.

Calculate, using the Central Limit Theorem, the smallest amount of money, rounded to the nearest 50 thousand, that the company must put into the fund.

• 750,000
• 850,000
• 1,050,000
• 1,150,000
• 1,400,000

An investor invests 100 dollars in a stock. Each month, the investment has probability 0.5 of increasing by 1.10 dollars and probability 0.5 of decreasing by 0.90 dollars. The changes in price in different months are mutually independent.

Calculate the probability that the investment has a value greater than 91 dollars at the end of month 100.

• 0.63
• 0.75
• 0.82
• 0.94
• 0.97

A city has just added 100 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions:

1. Each new recruit has a 0.4 probability of remaining with the police force until retirement.
2. Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.25.
3. The events of different new hires reaching retirement and the events of different new hires being married at retirement are all mutually independent events.

Calculate the probability that the city will provide at most 90 pensions to the 100 new hires and their husbands.

• 0.60
• 0.67
• 0.75
• 0.93
• 0.99

The minimum force required to break a particular type of cable is normally distributed with mean 12,432 and standard deviation 25. A random sample of 400 cables of this type is selected. Calculate the probability that at least 349 of the selected cables will not break under a force of 12,400.

• 0.62
• 0.67
• 0.92
• 0.97
• 1.00

An insurance company issues 1250 vision care insurance policies. The number of claims filed by a policyholder under a vision care insurance policy during one year is a Poisson random variable with mean 2. Assume the numbers of claims filed by different policyholders are mutually independent.

Calculate the approximate probability that there is a total of between 2450 and 2600 claims during a one-year period.

• 0.68
• 0.82
• 0.87
• 0.95
• 1.00

A charity receives 2025 contributions. Contributions are assumed to be mutually independent and identically distributed with mean 3125 and standard deviation 250.

Calculate the approximate 90th percentile for the distribution of the total contributions received.

• 6,328,000
• 6,338,000
• 6,343,000
• 6,784,000
• 6,977,000