Probability flashcards

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Sample space
The set of all possible outcomes of an experiment.
Event
A set of outcomes from the sample space, e.g. “the score is even”.
Complement A′
The event that A does not happen: every outcome not in A.
Mutually exclusive events
Events that cannot happen together: P(A ∩ B) = 0.
Independent events
The occurrence of one does not change the probability of the other: P(A ∩ B) = P(A) × P(B).
Conditional probability P(A | B)
The probability that A happens, given that B is known to have happened.
Exhaustive events
Events that between them cover the whole sample space, so P(A ∪ B) = 1.
Probability with equally likely outcomes
P(A) = n(A) / n(S) (No — learn it)
Complement rule
P(A′) = 1 − P(A) (No — learn it)
Addition law
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (Yes)
Multiplication law
P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A) (Yes)
Conditional probability formula
P(A | B) = P(A ∩ B) / P(B) (Yes, rearranged from the multiplication law)
Test for independence
P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A) (No — learn it)
Two-event Venn diagram: what goes in the region inside A but outside B?
P(A ∩ B′) = P(A) − P(A ∩ B), not P(A).
Order for filling a three-event Venn diagram
Centre (all three) first, then each “exactly two” region, then each “one only” region, then the outside.
Tree diagram: what do the second-stage branches show?
Conditional probabilities: the probability of the second outcome given the first.
Tree diagram: multiply or add?
Multiply along a path; add the probabilities of different paths that make up the event.
Two-stage experiment with dice or spinners
Draw a grid: one stage across, one down. Each cell is one equally likely outcome.
Independence pictured as area
In a unit square, A a vertical strip and B a horizontal strip: the overlap is a rectangle of area P(A) × P(B).
Why can't two events with non-zero probabilities be both mutually exclusive and independent?
Mutually exclusive gives P(A ∩ B) = 0, but independence needs P(A ∩ B) = P(A)P(B) > 0.
Why is P(A | B) usually different from P(B | A)?
Both have numerator P(A ∩ B), but one divides by P(B) and the other by P(A).
How does “without replacement” change a tree?
On later branches the total falls by 1 and the count of the item already taken falls by 1; the stages are dependent.
If A and B are independent, what about A and B′?
They are independent too, as are A′ and B, and A′ and B′.
Reversing a conditional probability on a tree
P(first | second) = (path through both) ÷ (sum of all paths giving the second outcome).
Two fair dice. P(at least one six)?
1 − (5/6)² = 11/36.
P(A) = 0.45, P(B) = 0.3, P(A ∪ B) = 0.6. Find P(A ∩ B) and P(neither).
P(A ∩ B) = 0.45 + 0.3 − 0.6 = 0.15; P(neither) = 1 − 0.6 = 0.4.
P(A) = 0.4, P(A ∪ B) = 0.7, P(B) = p. Find p if A, B are (i) mutually exclusive (ii) independent.
(i) 0.4 + p = 0.7, p = 0.3. (ii) 0.4 + p − 0.4p = 0.7, 0.6p = 0.3, p = 0.5.
6 lemon and 4 orange sweets; two eaten one after the other. P(same flavour)?
6/10 × 5/9 + 4/10 × 3/9 = 30/90 + 12/90 = 7/15.
Seeds: 60% from A (germinate 0.9), 40% from B (germinate 0.75). P(B | seed fails)?
P(fail) = 0.06 + 0.1 = 0.16; P(B | fail) = 0.1/0.16 = 0.625.
Of 160 adults, 72 hired a guide; of 250 visitors, 90 hired a guide. Are “adult” and “hired a guide” independent?
P(guide | adult) = 72/160 = 0.45 but P(guide) = 90/250 = 0.36: different, so not independent.
P(A) = 0.2, P(B | A) = 0.6, P(B | A′) = 0.3. Find P(B) and P(A | B).
P(B) = 0.12 + 0.24 = 0.36; P(A | B) = 0.12/0.36 = 1/3.
Venn counts: n(C) = 56, n(S ∩ C) = 20. P(S | C)?
Divide by the given event: 20/56 = 5/14.
“Given that” in a question
The event after “given that” is the denominator: its probability or count becomes the new total.
Showing independence
Calculate P(A ∩ B) and P(A) × P(B), write them side by side, and state the conclusion in words.
“At least one” probabilities
Use 1 − P(none); adding single-event probabilities double-counts overlaps.
Checking a Venn diagram or tree
The regions (or the branches from each point) must add to 1, or to the survey total for counts.
Accuracy of probability answers
Give an exact fraction or exact decimal; if rounding, 3 significant figures, with full accuracy kept in the working.

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