Probability of Annual Profit in Insurance Company

Health Insurance Policy

• An insurer reimburses 100% of health care costs between 20 and 120.
• Health care costs above 120 are reimbursed at 50%.
• The cumulative distribution function of reimbursements is G.

Claim Submissions

• The joint probability function of the number of claims submitted in April and May is given by a formula involving exponential functions.
• The expected number of claims submitted in May, given that exactly 2 claims were submitted in April, is a function of e.

Probability of Defective Modems

• A store has 80 modems from two sources, A and B, with 30 from A and 50 from B.
• The probability of a defective modem from A is 20% and from B is 8%.
• The probability that exactly two out of a sample of five modems are defective is calculated.

Life Insurance Policy

• A policy pays 5000 if the policyholder dies before age 50 and 0 otherwise.
• The probability that a policyholder will file more than three claims during a year is calculated.

Automobile Losses

• The amounts of automobile losses are mutually independent and uniformly distributed between 0 and 20,000.
• The insurer covers each loss subject to a deductible of 5,000.
• The probability that the total payout on 200 reported losses is between 1,000,000 and 1,200,000 is calculated.

Insurance Profile

• An insurance agent offers auto, homeowners, and renters insurance.
• The purchase of homeowners and renters insurance is mutually exclusive.
• The profile of the agent's clients is given by probabilities of having different products.
• The percentage of clients with both auto and renters insurance is calculated.

Cumulative Distribution Function

• The cumulative distribution function for health care costs is modeled by a function involving exponential functions.
• The policy has a deductible of 20.
• A random variable X has a cumulative distribution function involving a different formula.
• The expected value of the largest of three claims is calculated.

Charity Contributions

• A charity receives 2025 contributions with a mean of 3125 and standard deviation of 250.
• The approximate 90th percentile for the distribution of the total contributions is calculated.

Claims Distribution

• Claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000.
• The probability that the average of 25 randomly selected claims exceeds 20,000 is calculated.

Poisson Distribution

• The number of claims filed by a policyholder under a vision care insurance policy is a Poisson random variable with mean 2.
• The approximate probability that there is a total of between 2450 and 2600 claims during a one-year period is calculated.

Light Bulb Lifetime

• The lifetime of a light bulb is normally distributed with mean 3 and variance 1.
• The probability that at least one light bulb lasts for more than 3 months is calculated.

Independent Events

• Four events are given, and the probability that their union equals the sum of their probabilities is calculated.

Envelope Probability

• Four letters to different insureds are prepared along with accompanying envelopes.
• The probability that at least one letter ends up in its accompanying envelope is calculated.

Health Insurance Policy

• A health insurance policy covers visits to a doctor's office with a cost of 100.
• The annual deductible is 350.
• The probability that a policyholder had exactly five office visits, given that costs exceed the deductible, is calculated.

Machine Parts

• A machine has two parts, A and B, with different probabilities of working for one year.
• The probability that at least one part works for one year is given.
• The probability that part B works for one year, given that part A works for one year, is calculated.

Profit Distribution

• The annual profit of a life insurance company is normally distributed.
• The probability that the annual profit does not exceed 2000 and 3000 is given.
• The probability that the annual profit does not exceed 1000 is calculated.

Automobile Claim

• The loss on an automobile claim is a random variable with a cumulative distribution function involving a formula.
• The probability that the loss is between 0.5V and 1.0V is calculated.

Independent Events

• Two events, E and F, are independent with given probabilities.
• The probability that exactly one of the two events occurs is calculated.

Flood Insurance

• The number of claims received in a month is a random variable with a given probability distribution.
• The probability that more than three claims will be received during a consecutive two-month period, given that fewer than two claims were received in the first month, is calculated.