The following is known about the monthly claim frequency:

• The monthly claim frequency has a geometric distribution
• The probability that there are 4 claims in a month is half the probability that there are 2 claims in a month
• Monthly claim frequencies are independent

Determine the variance of the annual claim frequency.

• 3.41
• 38.95
• 40.97
• 42.15
• 69.94

A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study drop out before the end of the study with probability 0.2 (independently of the other participants).

Calculate the probability that at least nine participants complete the study in one of the two groups, but not in both groups.

• 0.096
• 0.192
• 0.235
• 0.376
• 0.469

Each time a hurricane arrives, a new home has a 0.4 probability of experiencing damage. The occurrences of damage in different hurricanes are mutually independent.

Calculate the mode of the number of hurricanes it takes for the home to experience damage from two hurricanes.

• 2
• 3
• 4
• 5
• 6

A company has five employees on its health insurance plan. Each year, each employee independently has an 80% probability of no hospital admissions. If an employee requires one or more hospital admissions, the number of admissions is modeled by a geometric distribution with a mean of 1.50. The numbers of hospital admissions of different employees are mutually independent. Each hospital admission costs 20,000.

Calculate the probability that the company’s total hospital costs in a year are less than 50,000.

• 0.41
• 0.46
• 0.58
• 0.69
• 0.78

On any given day, a certain machine has either no malfunctions or exactly one malfunction. The probability of malfunction on any given day is 0.40. Machine malfunctions on different days are mutually independent. Calculate the probability that the machine has its third malfunction on the fifth day, given that the machine has not had three malfunctions in the first three days.

• 0.064
• 0.138
• 0.148
• 0.23
• 0.246

The number of tornadoes in a given year follows a Poisson distribution with mean 3.

Calculate the variance of the number of tornadoes in a year given that at least one tornado occurs.

• 1.63
• 1.73
• 2.66
• 3.00
• 3.16

Under an insurance policy, a maximum of five claims may be filed per year by a policyholder. Let [math]p(n)[/math] be the probability that a policyholder files n claims during a given year, where [math]n = 0,1,2,3,4,5 [/math]. An actuary makes the following observations:

1. [math]p(n) ≥ p (n + 1) [/math] for [math]n = 0, 1, 2, 3, 4 [/math] .
2. The difference between [math]p (n) [/math] and [math]p(n + 1)[/math] is the same for [math] n = 0,1,2,3,4 [/math].
3. Exactly 40% of policyholders file fewer than two claims during a given year.

Calculate the probability that a random policyholder will file more than three claims during a given year.

• 0.14
• 0.16
• 0.27
• 0.29
• 0.33

An airline finds that 4 percent of the passengers that make reservations on a particular flight will not show up. Consequently, their policy is to sell 100 reserved seats on a plane that has only 98 seats. Find the probability that every person who shows up for the flight will find a seat available.

• 0.81
• 0.85
• 0.88
• 0.91
• 0.94

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

A manufactured lot of buggy whips has 20 items, of which 5 are defective. A random sample of 5 items is chosen to be inspected. Find the probability that the sample contains exactly one defective item if the sampling is done without replacement.

• 0.44
• 0.46
• 0.48
• 0.5
• 0.52

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Exactly one of six similar keys opens a certain door. If you try the keys, one after another, what is the expected number of keys that you will have to try before success?

• 2.5
• 3
• 3.5
• 4
• 4.5

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.