at most a given sum Sometimes you care about “sum at least 10” or “sum at most 4”. In that case: \(\mathbb{P}(S \le t) = \sum_{k=n}^{t} \mathbb{P}(S = k)\) \(\mathbb{P}(S \ge t) = \sum_{k=t}^{n s} \mathbb{P}(S = k)\) The calculator automatically sums over the relevant range for you and reports a single probability. Successes on dice and the binomial distribution Many games define a success as rolling certain faces (e.g. “5 or 6 on a d6”). If there are \(f\) success faces on an s-sided die, the per-die success probability is \(p = \dfrac{f}{s}\). If you roll \(n\) dice, each with success probability \(p\), the number of successes \(K\) follows a binomial distribution : \(\mathbb{P}(K = k) = \binom{n}{k} p^k (1 - p)^{n-k}\). Example: at least one 6 on 4d6 For a d6, one face is a 6, so \(p = 1/6\). For 4 dice, the probability of at least one 6 is \(\mathbb{P}(K \ge 1) = 1 - \mathbb{P}(K = 0) = 1 - (1 - p)^4 = 1 - (5/6)^4 \approx 51.77\%\). Exact probability vs odds It’s often helpful to express probability in different ways: Decimal: a number between 0 and 1, like 0.25. Percentage: the decimal times 100%, like 25%. Odds “1 in N”: \(N \approx 1

Calculators in at most a given sum Sometimes you care about “sum at least 10” or “sum at most 4”. In that case: \(\mathbb{P}(S \le t) = \sum_{k=n}^{t} \mathbb{P}(S = k)\) \(\mathbb{P}(S \ge t) = \sum_{k=t}^{n s} \mathbb{P}(S = k)\) The calculator automatically sums over the relevant range for you and reports a single probability. Successes on dice and the binomial distribution Many games define a success as rolling certain faces (e.g. “5 or 6 on a d6”). If there are \(f\) success faces on an s-sided die, the per-die success probability is \(p = \dfrac{f}{s}\). If you roll \(n\) dice, each with success probability \(p\), the number of successes \(K\) follows a binomial distribution : \(\mathbb{P}(K = k) = \binom{n}{k} p^k (1 - p)^{n-k}\). Example: at least one 6 on 4d6 For a d6, one face is a 6, so \(p = 1/6\). For 4 dice, the probability of at least one 6 is \(\mathbb{P}(K \ge 1) = 1 - \mathbb{P}(K = 0) = 1 - (1 - p)^4 = 1 - (5/6)^4 \approx 51.77\%\). Exact probability vs odds It’s often helpful to express probability in different ways: Decimal: a number between 0 and 1, like 0.25. Percentage: the decimal times 100%, like 25%. Odds “1 in N”: \(N \approx 1.