Compound probability plays a fundamental role in calculating insurance premiums, reserves and in defining corporate strategy. Compound probability is the probability that two independent events will occur simultaneously. It is the basis of probability distribution, which an actuary uses to evaluate insurable risks. In turn, risk estimations are used as guidelines when underwriters price policies and when corporate executive officers make decisions about new markets.
Two events are considered to be independent if the outcome of one has no impact on the output of another. For instance, rolling a die twice results in two independent outcomes. Coming up with a five on the first roll does not make the second roll any more or less likely to be a three, for example. In insurance, many events are assumed to be independent, such as the paths of two separate hurricanes or the likelihood that two different subscribers will have a car accident. In order to calculate the compound probability of two independent events, the probability of the first event is multiplied by the second event.
As an example, consider that two subscribers with life insurance policies at the same company will both die this year. The first, Harry, has a 20% chance of dying this year, and the second, Larry, has a 10% chance of dying this year. The compound probability that Harry and Larry will both die during the year is 0.10*0.20 = .02 or 2%. The fact that Harry dies has no effect on Larry dying and vice versa.
In contrast, the probability of dependent events occurring simultaneously is found using conditional probability. The occurrence of one dependent event will have an effect on the likelihood of another dependent event. For instance, rolling a five on the first roll of the dice will make it impossible for the sum of two rolls to be four. In casualty insurance, the likelihood of a fire in one subscriber’s house is increased if the next door neighbor’s house is on fire. Conditional probability is calculated using Bayes’ Theorem.
Insurance companies are very interested in analyzing the likelihood that any number of all their subscribers will make a claim during the year because they payout indemnities. An actuary uses compound and conditional probabilities to calculate the likelihood of payouts for each policy. Then an insurance underwriter will use the expected payout value to come up with a competitive premium rate.
Other actuaries use compound and conditional probabilities to analyze an insurance company's cash reserves to ensure the company will not go bankrupt paying out indemnities. Unlikely but not impossible scenarios must be considered, such as a category four hurricane hitting a major city or a terrorist attack in a major metropolitan area. Insurance company executives are interested in the evaluation of risks using compound probability when considering launching into developing markets as well.