Probability Calculator

Expected value

Probability also tells you what to expect on average. Expected value = Σ (outcome × probability). A game that pays ₹100 on a roll of six and nothing otherwise, costing ₹20 to play, has an expected value of 100 × 1/6 − 20 ≈ −₹3.33 per game. Over many plays the player loses about ₹3.33 each time, which is the basic arithmetic behind every casino and lottery.

Basic probability

For equally likely outcomes, probability = favourable outcomes ÷ total outcomes. Rolling two dice has 36 outcomes, and 6 of them sum to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so P(sum = 7) = 6/36 = 1/6 ≈ 16.7%. Probabilities range from 0 (impossible) to 1 (certain).

Combining events

For independent events, multiply: the chance of two heads in a row is 0.5 × 0.5 = 0.25. For either of two mutually exclusive events, add: P(rolling 1 or 2) = 1/6 + 1/6 = 1/3. If events can overlap, subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).

The complement rule is often the easiest route: P(at least one) = 1 − P(none). The chance of at least one six in four rolls is 1 − (5/6)⁴ ≈ 1 − 0.482 = 0.518, slightly better than even — a famous 17th-century gambling puzzle.

Common misunderstandings

Independent events have no memory: after five tails in a row, the next fair coin toss is still 50/50. Odds are not the same as probability; odds of 1 to 5 mean a probability of 1/6. And small probabilities add up over many trials, which is why rare events happen surprisingly often in large populations.