One doesn't affect the other → multiply: Two events are independent if one happening doesn't change the other's probability.
Then P(A ∩ B) = P(A) × P(B) — multiply to get 'both happen'.
Independent events: the second branch probabilities match the first — one outcome doesn't change the next.
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IB-style question — both happen
P(A) = 0.6 and P(B) = 0.5, and A and B are independent.
Find P(A ∩ B).
Step by step
- Independent → multiply.
- Evaluate.
Final answer
P(A ∩ B) = 0.3.
Independent = multiply for 'and': For independent events, 'A and B' is a simple product — no tree needed.
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Check whether P(A∩B) = P(A)·P(B): To test independence, compare the actual P(A ∩ B) with the product P(A) × P(B).
If they're equal, the events are independent; if not, they're dependent.
IB-style question — are they independent?
P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2.
Determine whether A and B are independent.
Step by step
- Compute the product.
- Compare with the given intersection.
Final answer
Since P(A)·P(B) = 0.2 = P(A ∩ B), the events are independent.
Show the comparison: State both numbers and that they're equal (or not) — that comparison is what earns the conclusion mark.
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Two different ideas — don't confuse them: Mutually exclusive: can't both happen, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
Independent: one doesn't affect the other, so P(A ∩ B) = P(A)·P(B).
These are different properties.
Mutually exclusive
- can't both happen
- P(A ∩ B) = 0
- P(A ∪ B) = P(A) + P(B)
Independent
- one doesn't affect the other
- P(A ∩ B) = P(A)·P(B)
- usually a non-zero overlap
IB-style question — mutually exclusive union
A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.45.
Find P(A ∪ B).
Step by step
- Mutually exclusive → P(A ∩ B) = 0.
- Evaluate.
Final answer
P(A ∪ B) = 0.75.
Use independence inside the addition rule: When events are independent, substitute P(A ∩ B) = P(A)·P(B) into the addition rule to find a missing probability: P(A ∪ B) = P(A) + P(B) − P(A)·P(B).
IB-style question — find a missing probability
A and B are independent, with P(A) = 0.3 and P(A ∪ B) = 0.65.
Find P(B).
Step by step
- Addition rule with independence.
- Collect and solve.
Final answer
P(B) = 0.5.
Let P(B) be the unknown: Write P(A ∩ B) as P(A)·P(B), substitute, and solve the linear equation for the unknown probability.