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The law of large numbers an easy concept to grasp, but when you start doing the math, it becomes a little more difficult. The concept is as simple as a coin flip. Flip a coin and the odds are 50-50 that the coin will land heads. But the chances it will land heads two times in a row aren't what you might think. The same with three times or five times in a row. So with two equal and independent probabilities of either heads or tails on each flip, you might think each flip has a 50-50 chance. But when you get to flipping five heads in a row, the formula becomes ½ x ½ x ½ x ½ x ½ or 1/32. When the odds are so heavily weighted in a limited number of flips, heads becomes increasingly unlikely. But over 10,000 flips or 100,000 flips the law of large number takes over and, if not exact, the number of heads will come up very close to 50 percent of the time.
This becomes a lot more difficult to quantify. Back when the sailors were trying to handicap the likelihood of a ship's return with its cargo, it depended on the quality of the ship, the crew, the captain, the waters, the time of year and piracy, to name just a few variables. But here is where the numbers start to add up over time. How many times has a ship successfully returned from the venture? Record keeping became important then, as it is now. Again, the law of large numbers began to play a role as more trips were navigated. The record keeping determined a rough probability. Then they had to factor in another variable – the value of the cargo.
Most people are required to have car insurance, home insurance, mortgage insurance and others elect to buy health insurance and renters insurance,
for example. It's all based on the same things the sailors based their estimates on, only it has become much more complicated and specialized. While the law of large numbers helps protect against unforeseen events, shared risk requires large numbers to work. That's where insurance companies come into play.
Each type of insurance is different, but the basic principles are the same. Each insurance provider needs to establish actuarial likelihoods of events occurring for which they will have to pay to cover the costs. That requires that insurance companies establish probability distributions, which plot the possibilities of an event against its probability. The more limited the possibilities, the simpler it is to establish a premium. But where there is a continuous range of possibilities, an insurance company must determine a central tendency which, in math terms, is the sum of each possible event times the probability of that event occurring. The mean is equal to the sum of each amount of possible loss events times the probability of that loss. You don't need to know all the math or the actuarial tables involved in establishing the mean for each event but it is the principle that establishes the value of the premium you will pay for any insurance.
Back to Tossing a Coin
It is the law of large number that disperses the cost of the mean payments over a large base of premiums that pay for the loss. That's why there are insurance companies and not just you and your neighbor insuring each other. As the number of those insured grows, the distribution of bad events gets closer to the mean (the 50-50 coin toss) with a smaller deviation of error and closer to the normal probability of an insurable event occurring. It's insurance companies that put together such large numbers to minimize the risk to within a fraction of the mean, making premiums (plus costs and profit) a quantifiable sum.