Coin Flip & Dice Roll Simulator
Simulate coin flips and dice rolls to explore probability concepts
Enter 1-10000 trials (more trials = better approximation)
Theory & Formula
What is Probability Simulation?
Probability simulation uses random number generation to model real-world random events. By running many trials, we can estimate theoretical probabilities through experimental results.
Theoretical vs Experimental Probability
Theoretical probability is calculated using mathematical formulas. For a fair coin, the probability of heads is:
Heads: \(P(A) = \frac{1}{2} = 0.5\)Experimental probability is calculated from actual results of trials:
Here f is the number of favorable outcomes and n is the total number of trials. \(\widehat{p} = \frac{f}{n}\)Law of Large Numbers
As the number of trials increases, experimental probability approaches theoretical probability. This is why more trials give better approximations.
Dice Probability
For a single fair die, each number (1-6) has equal probability:
\(P(X = 6) = \frac{1}{6} \approx 0.167\)For multiple dice, some sums are more likely than others. For example, rolling two dice:
\(P(X_1 + X_2 = 7) = \frac{6}{36} = \frac{1}{6}\)