Question 1

How do I find the probability of two events both happening?

Accepted Answer

For independent events, multiply: P(A and B) = P(A) × P(B). A 50% chance and a 40% chance happen together 0.5 × 0.4 = 20% of the time. This is the special case of OpenStax’s general multiplication rule, P(A and B) = P(B) × P(A | B), when knowing B does not change the chance of A.

Question 2

What is the probability of either event happening?

Accepted Answer

P(A or B) = P(A) + P(B) − P(A and B), so you do not count the overlap twice: 50% + 40% − 20% = 70%. Only when the events cannot happen together, so that P(A and B) = 0, is it enough to add the two probabilities.

Question 3

What is the chance of something happening at least once?

Accepted Answer

One minus the chance it never happens: 1 − (1 − p)ⁿ. The chance of at least one six in four rolls of a die is 1 − (5/6)⁴ = 51.8%. For an event with a 30% chance, five tries give 1 − 0.7⁵ = 83.2%: more tries help, but never reach 100%.

Question 4

What does independent mean?

Accepted Answer

That one event does not change the other's chance — like two coin flips. For events that affect each other, such as drawing cards without putting them back, the calculator's answers do not apply. OpenStax’s test: A and B are independent if P(A and B) = P(A) × P(B), or equivalently if P(A | B) = P(A).

Question 5

How do I turn odds into a probability?

Accepted Answer

Odds of 3 to 1 against mean 1 success in 4 tries: a probability of 1 ÷ (3 + 1) = 25%.

Question 6

Worked example: everything the calculator shows

Accepted Answer

Let A have a 50% chance and B a 40% chance, independent. Both happen: 20%. Either happens: 70%. Only A happens: 30% (0.5 × 0.6). Only B happens: 20% (0.5 × 0.4). Neither happens: 30% (0.5 × 0.6). Exactly one happens: 50%. The four exclusive cases, both, only A, only B and neither, add up to 100%, which is a quick check on any set of results. For three independent events of 50%, 40% and 25%, the chance that at least one happens is 1 − 0.5 × 0.6 × 0.75 = 77.5%.

Question 7

Mutually exclusive is not the same as independent

Accepted Answer

Mutually exclusive events cannot happen together, so P(A and B) = 0. Independent events can happen together, and their joint chance is the product of the two. If both events have a positive chance, they cannot also be independent: knowing that one happened tells you the other did not, which makes them dependent. In OpenStax’s swim-team example, 28 of 150 swimmers are novices and 80 of 150 practice four times a week, so independence would predict (28/150) × (80/150) ≈ 0.0996 for a novice who practices four times; the actual figure is 10/150 ≈ 0.0667, so the events are not independent.

Question 8

Common mistakes with probability

Accepted Answer

First, adding probabilities for “or” without subtracting the overlap, which can push the total above 100%. Second, multiplying when the events are dependent, such as two cards from the same deck. Third, the gambler’s fallacy: after four heads in a row a fair coin still has a 50% chance of heads, because the flips are independent. Fourth, entering 0.5 in a field that expects a percentage: the fields take percentages, so type 50.

Probability calculator