Bingo Odds Calculator (Chance of a Line or Full Card After N Calls)

How it works

Chance that j given numbers are all called after n of N balls: P(j) = n!(N − j)! ÷ [(n − j)! N!] · full card = P(24) for 75-ball, P(15) for 90-ball · any line = inclusion–exclusion over the 12 lines of a 75-ball card (centre free) or the 3 rows of a 90-ball ticket: 3P(5) − 3P(10) + P(15) · several cards ≈ 1 − (1 − p)^cards (cards treated as independent) · expected calls for a full card = m(N + 1) ÷ (m + 1)

Bingo is a race to have all the numbers of a pattern called. For a single card the probability is exact: the chance that a set of j numbers has been drawn after n calls follows the hypergeometric distribution, and inclusion–exclusion combines the overlapping lines of a card (rows, columns and diagonals sharing numbers). A 75-ball card needs 24 numbers for a blackout, a 90-ball ticket 15 numbers for a full house, so a full card usually takes 70 to 85 calls. Several cards are treated as independent, a slight overestimate because cards in one game share the same draw.

Worked example

75-ball bingo, one card, 40 numbers called: each of the 12 lines needs 4 or 5 specific numbers, and combining them by inclusion–exclusion gives a 44.56% chance that at least one line is complete; on average one card needs 41.4 calls for a line. A blackout needs all 24 numbers among the 40 called, a probability of only 2.44 × 10^-7% (about 1 in 410 million); on average it takes 73.0 calls.

Sources: Wolfram MathWorld — Hypergeometric Distribution · Wizard of Odds — Bingo

Frequently asked questions