Understanding probability through the simple elegance of a coin flip. See the law of large numbers in action with our interactive simulator.
Probability is the mathematical study of randomness and uncertainty. A fair coin flip is the perfect introduction to probability because it has exactly two equally likely outcomes: heads or tails. Each outcome has a probability of exactly 50%, or 0.5, or 1/2.
But here's where it gets interesting: knowing the probability doesn't tell you what will happen on any single flip. You could flip heads 10 times in a row (probability: 0.1%), and the next flip would still be exactly 50% heads. Each flip is independent.
Watch how the percentage of heads approaches 50% as you flip more coins. This is the Law of Large Numbers in action.
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One of the most important principles in probability is the Law of Large Numbers, discovered by mathematician Jacob Bernoulli in 1713. It states that as you increase the number of trials, the average result will get closer to the expected value.
For coin flips, this means:
You might see 70% heads or 30% heads. Small sample sizes have high variance and can seem very "unfair."
The percentage will typically be between 40-60% heads. The distribution starts to stabilize.
You'll almost always see 47-53% heads. The more flips, the closer to 50% you'll get.
The percentage will be extremely close to 50.0%, typically within 0.1%. This is mathematical certainty in action.
One of the most common mistakes in probability is the Gambler's Fallacy - the belief that past random events affect future ones.
Wrong thinking: "I've flipped heads 5 times in a row, so tails is 'due' to come up next."
Correct thinking: "Each flip is independent. The probability of heads on the next flip is still exactly 50%, regardless of what came before."
The coin has no memory. It doesn't know or care what happened on previous flips. This is why casinos love roulette players who bet based on "hot" or "cold" numbers - they're falling for the Gambler's Fallacy.
Humans are pattern-seeking creatures, and we often see meaningful patterns in random data. But streaks are completely normal in random sequences:
If you flip a coin 1,000 times, you'll almost certainly see a streak of at least 8 of the same result. In 10,000 flips, you'd expect to see a streak of 12 or more. These aren't anomalies - they're mathematically expected!
For a perfect mathematical coin, yes. For physical coins, there's typically a slight bias (usually 51/49 or better) due to the weight distribution. Digital coin flips using cryptographic random number generators, like FlipCoin, are actually more fair than physical coins because they have true 50/50 probability.
No. Each flip is an independent event with 50% probability for each outcome. No amount of analyzing previous results, looking for patterns, or using any system can predict or influence the next result. This is fundamental to how randomness works.
Technically possible but incredibly rare. Research suggests the probability is approximately 1 in 6,000 for a US nickel spun on a hard surface. When flipped in the air and caught, it's essentially zero. Most coin flip rules treat edge-landing as a re-flip.
Because of the Law of Large Numbers combined with a built-in house edge. In roulette, for example, the true probability of winning a red/black bet is 47.4% (not 50%) due to the green zeros. Over millions of spins, the casino's edge guarantees profit, even though individual players might win short-term.
The probability is (1/2)^100, which is approximately 1 in 1.27 x 10^30 (that's a 1 followed by 30 zeros). To put this in perspective, if every person on Earth flipped a coin once per second since the Big Bang, we still wouldn't expect to see this happen even once.
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