School Probability Problems: Coins, Dice, Independent Events, Outcome Tree — Online Solver

Choose a problem type, enter the data or take a ready-made wording, and get the probability as a fraction and a decimal together with a step-by-step solution.

How to solve school probability problems

Most exam problems reduce to the classical formula P = m / n: m is the number of favourable outcomes and n the number of all equally likely outcomes. The key is to count m and n correctly and to make sure the outcomes really are equally likely.

Coins and dice

n coin tosses give 2ⁿ outcomes, and exactly k heads occur in C(n; k) of them. Two dice give 36 outcomes, three dice 216. For sums a table helps: for two dice the sum 8 can be obtained in five ways (2+6, 3+5, 4+4, 5+3, 6+2), so the probability is 5/36.

Independent events and «at least one»

For independent events the probability that both happen is the product: P(A and B) = P(A)·P(B). The probability of «at least one» is easiest through the opposite event: P = 1 − (1 − p₁)(1 − p₂)·… For incompatible events (they cannot occur together) the probabilities simply add.

A series of shots and the tree

If a shooter hits with probability p, a miss has probability q = 1 − p, and the probability of a given series is the product of the probabilities of each shot. When there are many cases, draw a tree: the probability of a path is the product along its branches, and the probability of an event is the sum over suitable paths.

Writing and rounding the answer

Exam answers are decimals or integers. If the fraction terminates (3/8 = 0.375), write it exactly. If it repeats (5/36 = 0.1388…), round as the problem asks, usually to hundredths: 0.14. A probability is always between 0 and 1: if you get a number above 1 or a negative one, look for a mistake.

FAQ

How do I solve an «at least one» problem?

Find the probability of the opposite event «none happens» (multiply the probabilities that each event does not happen) and subtract it from one: P = 1 − q1·q2·…. For two shots with probabilities 0.8 and 0.9 this gives 1 − 0.2·0.1 = 0.98.

When do I multiply probabilities and when do I add them?

Multiply for independent events that happen together («and»): this first, then that. Add for incompatible cases of which any one will do («or»): either one or the other, but never both at once.

How many decimals should the answer have?

Follow the statement. If it says «round to hundredths», give two decimals. If the fraction terminates and nothing about rounding is said, write it exactly. The calculator shows the exact value as well as rounding to hundredths and thousandths.

Why do two dice give 36 outcomes and not 21?

The dice are distinguishable (first and second), so the pairs 1–2 and 2–1 are different equally likely outcomes. The sum 7 occurs in six ways out of 36, not once among 21 unordered pairs, and its probability is 1/6.

How do I find the probability of exactly k hits in n shots?

Choose the places of the hits: C(n; k) ways. Each way has probability p^k·(1 − p)^(n − k), so P = C(n; k)·p^k·(1 − p)^(n − k). This is the Bernoulli formula; for stages with different probabilities use the outcome tree.