Law of Large Numbers Statistics Example 3
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Example 3
mediumA basketball player has a career free-throw percentage of 80%. She misses 5 free throws in a row in one game. A commentator says she is 'due' to make the next one. Is this reasoning correct? Explain using the law of large numbers.
Solution
- 1 Step 1: The commentator's reasoning is the gambler's fallacy — the mistaken belief that past outcomes affect future independent events. Each free throw is independent; missing 5 in a row does not increase the probability of making the next one.
- 2 Step 2: The law of large numbers says that over many thousands of free throws, her proportion of makes will approach 80%. It does NOT say that short-run deviations must be immediately corrected. She is not 'due' — each attempt still has an 80% success probability.
Answer
The commentator's reasoning is incorrect (gambler's fallacy). The law of large numbers applies to long-run averages, not short sequences. Each free throw has the same 80% probability regardless of recent results.
The gambler's fallacy misapplies the law of large numbers to short sequences. The law guarantees convergence of averages over very large samples, not correction of short-run streaks. Independent events have no memory — past outcomes do not influence future probabilities.
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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