Law of Large Numbers Statistics Example 4
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Example 4
hardAn insurance company insures 100,000 homeowners. The probability of a claim in a given year is 0.02, with an average claim of \50,000. They charge \1,200 per policy. (a) What is the expected number of claims? (b) What is the expected total payout? (c) What is the expected profit? (d) Explain why this business model works using the law of large numbers.
Solution
- 1 Step 1: (a) Expected claims: . (b) Expected payout: 2{,}000 \times \50{,}000 = \. (c) Revenue: 100{,}000 \times \1{,}200 = \. Expected profit: \120M - \100M = \20{,}000{,}000$.
- 2 Step 2: (d) With 100,000 policies, the actual number of claims will be very close to the expected 2,000 by the law of large numbers. This predictability allows the company to set premiums that reliably cover payouts with a margin. A single homeowner faces unpredictable risk, but the insurer's average payout per policy is highly predictable.
Answer
(a) 2,000 expected claims. (b) \100M expected payout. (c) \20M expected profit. (d) With 100,000 policies, the law of large numbers makes the average claim rate predictable, enabling reliable profit calculation.
Insurance is a practical application of the law of large numbers. Individual outcomes are unpredictable, but the average across many independent policies converges to the expected value. This allows insurers to set premiums that cover expected costs plus a profit margin with high confidence.
About Law of Large Numbers
The Law of Large Numbers states that as the number of independent, identically distributed trials increases, the sample average converges to the theoretical expected value (population mean). In other words, larger samples produce more reliable estimates of the true probability or average.
Learn more about Law of Large Numbers →More Law of Large Numbers Examples
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