The definition
P(event) = favourable outcomes ÷ total equally likely outcomes, always between 0 and 1.
Standard sample spaces
Experiment | Outcomes |
|---|---|
one coin | 2 |
two coins | 4 (HH, HT, TH, TT) |
three coins | 8 |
one die | 6 |
two dice | 36 ordered pairs |
a pack of cards | 52 (4 suits of 13; 26 red, 26 black; 12 face cards) |
Two dice. Sums of 7 come up most (6 ways). A sum of 9 has 4 ways: (3, 6), (4, 5), (5, 4), (6, 3), so P = 4/36 = 1/9.
At least one: use the complement
P(at least one) = 1 − P(none).
Example. Two coins: P(at least one head) = 1 − 1/4 = 3/4.
Independent events
If A and B don't affect each other, P(A and B) = P(A) × P(B).
Example. P(A) = 0.4 and P(B) = 0.5, so P(both) = 0.2.
Drawing without replacement
Multiply the probabilities step by step, reducing the counts each time.
Example. A bag holds 5 red and 3 blue balls. Two are drawn without replacement. P(both red) = 5/8 × 4/7 = 5/14. The same result comes from ⁵C₂ / ⁸C₂ = 10/28.
Counting with combinations
P(exactly two heads in three tosses) = ³C₂ / 8 = 3/8.
Odds
Odds in favour of an event of a : b mean P = a / (a + b). Odds in favour 3 : 5 give P = 3/8.
Calendar classic
A leap year has 366 days = 52 weeks + 2 days. Those 2 extra consecutive days are one of 7 pairs (Sun–Mon, Mon–Tue, …), and 2 of the pairs include Sunday. So P(53 Sundays) = 2/7.
Shortcuts and tricks
The standard way is to list the sample space and count. These shortcuts count without listing.
"At least one" by the complement
Example: three coins. P(at least one head)?
Standard way: list the 8 outcomes; all but TTT contain a head, so 7/8.
Shortcut: 1 − P(no heads) = 1 − 1/8 = 7/8.
Two-dice sums without a table
Example: P(sum = 8) with two dice.
Standard way: list (2, 6), (3, 5), (4, 4), (5, 3), (6, 2): 5 of 36.
Shortcut: a sum s up to 7 has s − 1 ways, and above 7 it has 13 − s ways. Sum 8 has 5 ways: 5/36.
Exactly k heads in n tosses: nCk/2ⁿ
Example: exactly 2 heads in 4 tosses.
Standard way: list all 16 outcomes; 6 have exactly two heads, so 6/16 = 3/8.
Shortcut: ⁴C₂/2⁴ = 6/16 = 3/8.
Drawing several at once
Example: 4 red and 6 blue balls; two drawn together. P(both blue)?
Standard way: 6/10 × 5/9 = 30/90 = 1/3.
Shortcut: ⁶C₂/¹⁰C₂ = 15/45 = 1/3. This scales better to three or more balls.
Exam tips
Make sure the outcomes you count are equally likely: (4, 5) and (5, 4) are different outcomes with two dice.
"Or" for mutually exclusive events means add; "and" for independent events means multiply.
Probabilities above 1 or below 0 mean a counting mistake.