Probability

Probability Rules

Applying the addition rule and multiplication rule to find probabilities.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Combining Probabilities of Two Events

A standard deck has 52 cards: 4 suits of 13 each, including 4 Kings and 4 Queens. Before reading on: would finding the probability of drawing a King or a Queen work the same way as finding the probability of drawing a King or a Heart?

Definition — Addition and Multiplication Rules

For mutually exclusive events (they can't both happen), P(A or B) = P(A) + P(B). For events that can overlap, the general addition rule subtracts the double-counted overlap: P(A or B) = P(A) + P(B) − P(A and B). For independent events, the multiplication rule gives P(A and B) = P(A) × P(B).

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Worked Example — Mutually Exclusive vs. Overlapping Events

P(King or Queen): these can't both happen on one draw (mutually exclusive), so P(King or Queen) = 4/52 + 4/52 = 8/52 = 2/13 ≈ 0.154. P(King or Heart): these can overlap (the King of Hearts is both), so the general rule is needed: P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 ≈ 0.308. Without subtracting the overlap, the King of Hearts would be counted twice.

Worked Example — The Multiplication Rule for Independent Events

Drawing a card, replacing it, then drawing again: P(King on the first draw AND King on the second draw) = P(King) × P(King) = (4/52) × (4/52) = (1/13) × (1/13) = 1/169 ≈ 0.006. Replacing the card between draws keeps the two draws independent — the deck's composition is identical each time.

Probability Simulator

Experiment

Type

Sides

Number of dice

Run a trial to see the outcome here.

Simulation

Target trials

Trials per tick

Track outcome

0 / 200 trials

Histogram — experimental vs theoretical

Running probability — Law of Large Numbers

Statistics

Mean

— experimental

3.5 theoretical

Variance

— experimental

2.917 theoretical

Standard deviation

— experimental

1.708 theoretical

Trials

0 experimental

— theoretical

Frequency table

Outcome	Experimental	Theoretical
1	—	0.167
2	—	0.167
3	—	0.167
4	—	0.167
5	—	0.167
6	—	0.167

Central Limit Theorem

Draws 300 independent samples of 5trials each from the current experiment, averages each sample, and histograms the resulting sample means — the distribution should look progressively more bell-shaped (normal) as the sample size grows, even though the experiment’s own distribution usually isn’t.

Sample size

Number of samples

Tip

Before applying the addition rule, always check whether the two events can happen together — the general rule (subtracting the overlap) always works, but the simpler mutually-exclusive version only applies when they genuinely can't overlap.

Common Mistakes

Using P(A) + P(B) for events that can actually overlap.
King and Heart can both be true at once (the King of Hearts) — using the simple addition rule here would double-count that card; the general rule's subtraction is required.
Applying the independent-events multiplication rule to draws made without replacement.
Without replacement, the deck's composition changes after the first draw, so the two draws aren't independent — a different (conditional) calculation is needed instead, covered in the next lesson.

Key Takeaways

P(A or B) = P(A) + P(B) only when A and B are mutually exclusive; otherwise the overlap P(A and B) must be subtracted.
P(A and B) = P(A) × P(B) only when A and B are independent.
Sampling with replacement preserves independence between draws; sampling without replacement does not.

Summary

The next lesson looks more closely at what independence actually means, and how to find a probability conditioned on another event already having happened.

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