Coin and Dice Probability Calculator

Frequently Asked Questions

What is the probability of getting heads every time in 10 flips?

P(X = 10) = 0.5<sup>10</sup> = 1/1024 ≈ 0.098%. It is possible but very unlikely, occurring roughly once every 1,024 attempts.

Why is rolling a sum of 7 with 2d6 the most likely outcome?

There are 6 ways to make a sum of 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) out of 36 total outcomes, giving a probability of 6/36 ≈ 16.67%. Every other sum has fewer combinations.

Does this assume independent rolls?

Yes. Each flip or die roll is assumed to be independent of all others. The result of one flip does not affect the next. This is the standard model for ideal random devices.

Can I compute the probability for 100 coin flips?

Yes. The iterative PMF method used here is stable for large n. Results for very large n may accumulate tiny floating-point rounding errors, but the three-decimal display is accurate in the practical range.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

This calculator has two modes. In coin mode, it uses the binomial distribution to find the probability of getting exactly k heads in n fair flips, at least k heads, and at most k heads. Because the binomial coefficient C(n,k) overflows JavaScript's floating-point representation for n > 170 when computed via factorial multiplication, the PMF is built iteratively: pmf[0] = 0.5ⁿ and each subsequent term uses the recurrence pmf[i+1] = pmf[i] × (n−i)/(i+1). This remains numerically stable for large n.

In dice mode, the full sum distribution is built by convolution: starting from a distribution on {0}, each die is added by shifting and combining. After rolling all dice, the probability of a target sum equals the count of favorable outcomes divided by sides^dice.

Coin Flip Formula

For n fair flips, the probability of exactly k heads is:

P(X = k) = C(n,k) × 0.5ⁿ

where C(n,k) = n! ÷ (k! × (n−k)!). For example, exactly 2 heads in 4 flips: C(4,2) = 6, so P = 6 × 0.5⁴ = 6/16 = 37.5%. The cumulative P(X ≤ k) sums this term from i = 0 to k; the tail P(X ≥ k) = 1 − P(X ≤ k−1).

Dice Sum: Why Middle Sums Are More Likely

For two six-sided dice (2d6), there are 6 × 6 = 36 equally likely outcomes. Only 1 combination sums to 2 (1+1), while 6 combinations sum to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). The distribution is triangular - shaped like a tent - with the most likely sum near the middle.

For three or more dice the distribution rapidly approaches a normal shape by the Central Limit Theorem. This is why casino games based on multiple dice are more predictable than single-die games: the central sums are far more probable than the extremes.

Common Mistakes

Asking for more heads than flips. The target k cannot exceed n; the probability would be 0.

Confusing "exactly k" with "at least k". P(X = 5) and P(X ≥ 5) are very different.

Forgetting the minimum dice sum equals the number of dice. You cannot roll less than 1 per die; with 3d6 the minimum is 3, not 0.

Assuming each sum is equally likely. Middle sums are far more probable than extremes for multi-die rolls.

Assumptions and FAQ

Are these fair coins and dice? Yes. Every outcome is treated as equally likely: each coin face has exactly 50% probability and each die face has probability 1/sides. This is the standard mathematical model; loaded dice or biased coins require different tools.

Why are percentages not round numbers? Many probabilities are fractions that do not divide evenly into decimal form. The calculator shows three decimal places to avoid misleading rounding.

Can I model 4-sided, 8-sided, or 20-sided dice? Yes. The dice mode supports d4, d6, d8, d10, d12, and d20, covering all standard tabletop RPG dice.

Coin and Dice Probability Calculator

Frequently Asked Questions

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