Joint probability calculator

The formulas

Independent: P(A ∩ B) = P(A) × P(B) Dependent: P(A ∩ B) = P(A) × P(B | A) From counts: P(A ∩ B) = n(A and B) ÷ n

The first is a special case of the second: when A and B are independent, P(B | A) = P(B). Once you have the joint probability, the addition rule gives the chance of at least one, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Worked example: independent events

Toss a coin and roll a die. What is the probability of heads (A) and a six (B)? The coin cannot affect the die, so multiply:

P(A ∩ B) = 1/2 × 1/6 = 1/12 = 0.0833333

The calculator also gives P(A ∪ B) = 0.5 + 0.166667 − 0.0833333 = 0.583333 and the chance of neither, (1/2)(5/6) = 0.416667. The Venn diagram shows the four regions, which add to 1.

Worked example: dependent events

Draw two cards from a deck without replacing the first. What is the chance both are aces? The first is an ace with probability 4/52. If it was, 3 aces remain among 51 cards, so P(B | A) = 3/51:

P(A ∩ B) = 4/52 × 3/51 = 12/2652 = 1/221 = 0.00452489

Treating the draws as independent would give (4/52)² = 1/169, which overstates the chance because it pretends the first ace goes back into the deck.

Worked example: a table of counts

In the table mode, 100 people were asked whether they own a car (A) and a bike (B): 45 own both, 15 only a car, 25 only a bike and 15 neither. Dividing by 100 gives the joint probability table: P(A ∩ B) = 0.45, with marginals P(A) = 0.6 and P(B) = 0.7. If the two were independent the joint probability would be 0.6 × 0.7 = 0.42. The observed 0.45 is a little higher, so in this sample car owners are slightly more likely than average to own a bike.

How to interpret the result

A joint probability is always at most the smaller of P(A) and P(B), because both events happening is a special case of either one happening. Very small joint probabilities are common when several conditions must all hold: the chance of two aces in a row is only 1 in 221 even though each ace on its own is 1 in 13. The “1 in …” figure in the results is often easier to picture than a long decimal.

Common mistakes

• Multiplying probabilities of dependent events as if they were independent. Use P(B | A) for the second factor.
• Adding instead of multiplying. P(A) + P(B) is part of the “or” rule, not the “and” rule.
• Joint probabilities larger than either marginal. P(A ∩ B) can never exceed P(A) or P(B); the calculator rejects inputs that break this.
• Confusing P(A ∩ B) with P(A | B). The joint probability divides by the whole sample, the conditional one only by B.