Probability • Topic 3 of 4

Mutually Exclusive & "Or"

For events joined by "or", you add — but you must subtract any overlap so it is not counted twice. The general rule is P(A or B) = P(A) + P(B) − P(A and B). Events are mutually exclusive when they cannot both happen at once (rolling a 2 or a 5 on one die; drawing a single card that is a king or a queen), and then the overlap is zero, so it simplifies to P(A) + P(B). The trap is applying the simple version when the events can co-occur: for one card, P(king or heart) is 4/52 + 13/52 − 1/52 = 16/52, because the king of hearts sits in both sets and would otherwise be double-counted. Ask one question — "can both happen together?" If no, add straight; if yes, add and subtract the overlap. Contrast this with "and", where you multiply: "or" adds probabilities, "and" multiplies them.

✅ Solved examples

1. A fair die is rolled. Probability of a 2 or a 5?
These cannot both occur on one roll, so mutually exclusive: 1/6 + 1/6 = 2/6 = 1/3.
2. One card is drawn from a deck. Probability it is a king or a queen?
Mutually exclusive (a card cannot be both): 4/52 + 4/52 = 8/52 = 2/13.
3. One card is drawn. Probability it is a king OR a heart?
These overlap at the king of hearts: 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
4. Quantitative Comparison. A die is rolled. Quantity A: P(even OR greater than 3). Quantity B: 5/6. Which is greater?
Even = {2,4,6}; greater than 3 = {4,5,6}; union = {2,4,5,6} = 4 outcomes, so P = 4/6 = 2/3. (Overlap {4,6} was double-counted before pruning.) 2/3 ≈ 0.667 < 5/6 ≈ 0.833. Quantity B is greater.

✏️ Practice — try these, take hints as needed

1. A die is rolled. Probability of a 1 or a 4?
Cannot both occur — mutually exclusive.
1/6 + 1/6.
Add.
1/3
2. A card is drawn. Probability it is a heart or a spade?
A card cannot be both suits — no overlap.
13/52 + 13/52.
Add and simplify.
1/2
3. A card is drawn. Probability it is a queen or a diamond?
They overlap at the queen of diamonds.
4/52 + 13/52 − 1/52.
16/52.
4/13
4. A bag has 2 red, 3 blue, 5 green. Probability of red or blue in one draw?
Red and blue cannot both be the one ball drawn.
2/10 + 3/10.
Add.
1/2
5. A die is rolled. Probability of a number that is odd or greater than 4?
Odd = {1,3,5}; >4 = {5,6}.
Union = {1,3,5,6}.
4 of 6.
2/3

📝 Topic test — 8 questions

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