Coin Flip & Random Events Simulator

A Bernoulli trial is a random experiment with exactly two possible outcomes: success or failure. Each trial is independent, meaning the outcome of one trial doesn't affect others. Classic examples include flipping a coin (heads/tails), rolling a die and checking if it's a 6, or checking if a random person is left-handed. The probability of success (p) stays constant across all trials.

This fundamental theorem states that as you perform more trials, the average of your results tends to get closer to the expected (theoretical) value. With 10 coin flips, getting 70% heads isn't unusual. With 10,000 flips, you'll almost certainly be very close to 50%. This is why casinos are profitable despite individual players sometimes winning - over millions of bets, the house edge always wins.

When you repeat a Bernoulli trial n times and count the successes, the resulting distribution is called binomial. It tells you the probability of getting exactly k successes out of n trials. The histogram in this simulator compares your empirical results (what actually happened) with the theoretical binomial distribution (what probability predicts). As you run more experiments, the bars should align more closely with the curve.

A common misconception is that after a streak of one outcome, the opposite is "due." This is false! Each trial is independent - the coin doesn't remember its history. After 10 heads in a row, the probability of the next flip being heads is still exactly 50%. This simulator can help you see that streaks are normal parts of randomness, not signs that something is about to change.