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Independent Events Probability Calculator

All the and/or probabilities at once.

How it is calculated

Enter P(A) and P(B) as percentages from 0 to 100. The tool converts them to probabilities (divide by 100) and assumes the two events are independent. It then reports P(A and B) = P(A) × P(B), P(A or B) = P(A) + P(B) − P(A) × P(B) and P(neither) = (1 − P(A)) × (1 − P(B)). The three results are percentages with 2 decimal places. The subtraction in the second formula removes the overlap that would otherwise be counted twice. Values below 0 or above 100, and empty boxes, give no result. A probability of 0 or 100 is allowed. P(A or B) and P(neither) always add up to 100 %. Independence is an assumption you must justify; the tool cannot check it.

Worked example

Let P(A) = 30 % and P(B) = 40 %, so 0.30 and 0.40. Both happen: 0.30 × 0.40 = 0.12, which the tool prints as 12.00%. At least one happens: 0.30 + 0.40 − 0.12 = 0.58, so 58.00%. Neither happens: 0.70 × 0.60 = 0.42, so 42.00%. As a check, 58.00% + 42.00% = 100 %, because either at least one event happens or neither does.

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Common mistakes

  • Typing 0.3 instead of 30. The boxes expect percentages, so 0.3 is read as 0.3 %, and every result shrinks accordingly.
  • Applying the tool to events that are not independent, such as drawing two cards from one deck without replacement. The first draw changes the second probability, so the formulas give wrong answers.
  • Confusing independent with mutually exclusive. Mutually exclusive events cannot both happen, so P(A and B) would be 0; here it is the product of the two.
  • Simply adding P(A) + P(B) for the chance of either. That counts the overlap twice; 30 % + 40 % = 70 %, but the correct answer is 58 %.
  • Expecting P(neither) to equal 100 % − P(A) − P(B). That would give 30 %, but the independent answer is 42 %.

About this calculator

Two events are statistically independent when the outcome of one has no effect on the probability of the other — flipping a coin twice, or rolling a die and drawing a card from a separate deck, are classic examples. For independent events A and B, the multiplication rule gives P(A and B) = P(A) × P(B); the addition rule gives P(A or B) = P(A) + P(B) − P(A) × P(B) to avoid double-counting the overlap; and the probability that neither happens is P(not A) × P(not B), since independence of A and B also implies independence of their complements.

These formulas underpin risk analysis (the odds that two unrelated system components both fail), quality control (the chance that two independently inspected batches both pass), genetics (probability that a child inherits two independently-assorting traits), and everyday reasoning about compound events like the odds of it raining and your flight being delayed on the same day. Confusing independent events with mutually exclusive ones is a common mistake — mutually exclusive events can never both happen, so P(A and B) = 0, which is the opposite situation from independence.

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