Probability with equally likely outcomes
When a fair coin is tossed, head and tail are equally likely, so each has a 1 in 2 chance.
If all outcomes of a sample space are equally likely, the probability of an event E is the number of favourable outcomes divided by the total number of outcomes. It is always between 0 and 1.
Formula: P(E) = n(E) ÷ n(S)
Rule 1: Dice and coins — Count favourable and total.
- Easy: Die, prime number: 3 out of 6, P = 1/2.
- Medium: Two coins, two heads: 1 out of 4, P = 1/4.
- Exam-type: Two dice, sum 7: 6 out of 36, P = 1/6.
Rule 2: Cards — 52 cards, 13 in each of 4 suits, 12 face cards, 26 red.
- Easy: P(king) = 4/52 = 1/13.
- Medium: P(heart) = 13/52 = 1/4.
- Exam-type: P(red face card) = 6/52 = 3/26.
Probability never exceeds 1 or goes below 0.
Using this formula when the outcomes are not equally likely.
Wanted over all.
Practice questions
- 1A die is rolled. Find P(prime number).
- 2A die is rolled. Find P(number greater than 4).
- 3A coin is tossed. Find P(head).
- 4Two coins are tossed. Find P(two heads).
- 5Two dice are rolled. Find P(sum is 7).
- 6Two dice are rolled. Find P(sum is 10).
- 7Three coins are tossed. Find P(exactly two heads).
- 8A card is drawn from a pack of 52. Find P(red face card).
Show answers (try them yourself first)
- 11/2
- 21/3
- 31/2
- 41/4
- 51/6
- 61/12
- 73/8
- 83/26
Probability of "not"
If the chance of rain is 2 in 5, the chance of no rain is the remaining 3 in 5.
The probability that A does not happen is 1 minus the probability that it does. This is useful for "at least one".
Formula: P(A′) = 1 − P(A)
Rule 1: Direct — Subtract from 1.
- Easy: P(A) = 0.3: P(not A) = 0.7.
- Medium: P(six) = 1/6: P(not six) = 5/6.
- Exam-type: P(rain) = 2/5: P(no rain) = 3/5.
Rule 2: At least one — At least one = 1 minus none.
- Easy: Three coins, at least one head: 1 − 1/8 = 7/8.
- Medium: Two dice, at least one six: 1 − 25/36 = 11/36.
- Exam-type: Two dice, sum at least 3: 1 − 1/36 = 35/36.
When a question says "at least", try 1 minus "none".
Counting "at least one head" case by case and missing outcomes.
At least one = 1 minus none.
Practice questions
- 1P(A) = 0.3. Find P(not A).
- 2A die is rolled. Find P(not a six).
- 3The chance of rain is 2/5. Find the chance of no rain.
- 4Two dice are rolled. Find P(sum is not 7).
- 5Two dice are rolled. Find P(sum is at least 3).
- 6Three coins are tossed. Find P(at least one head).
- 7Two dice are rolled. Find P(at least one six).
- 8P(A) = P(not A). Find P(A).
Show answers (try them yourself first)
- 10.7
- 25/6
- 33/5
- 45/6
- 535/36
- 67/8
- 711/36
- 81/2
Addition rule
To count students who play cricket or football, we must not count those who play both twice.
For any events A and B: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, P(A and B) = 0, so it becomes P(A) + P(B).
Formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Rule 1: Overlapping — Subtract the overlap once.
- Easy: Die: even or prime = {2, 3, 4, 5, 6}, P = 5/6.
- Medium: P(A) = 0.5, P(B) = 0.3, P(A and B) = 0.1: P(A or B) = 0.7.
- Exam-type: Card: king or heart = 4 + 13 − 1 = 16 cards, P = 4/13.
Rule 2: Mutually exclusive — No overlap to subtract.
- Easy: P(A) = 0.2, P(B) = 0.35 (exclusive): P(A or B) = 0.55.
- Medium: Die: 1 or 6: 1/6 + 1/6 = 1/3.
- Exam-type: Die: multiple of 2 or of 3: {2, 3, 4, 6}, P = 2/3.
Subtract the overlap, or the common outcomes are counted twice.
Adding the two probabilities without subtracting the common part.
Add both, remove the double count.
Practice questions
- 1A die is rolled. Find P(even or prime).
- 2P(A) = 0.5, P(B) = 0.3, P(A and B) = 0.1. Find P(A or B).
- 3A and B are mutually exclusive with P(A) = 0.2 and P(B) = 0.35. Find P(A or B).
- 4A card is drawn from 52. Find P(king or heart).
- 5P(A or B) = 0.8, P(A) = 0.5, P(B) = 0.6. Find P(A and B).
- 6A die is rolled. Find P(multiple of 2 or multiple of 3).
- 7A card is drawn from 52. Find P(heart or face card).
- 8P(A) = 0.4, P(B) = 0.5, P(A or B) = 0.7. Find P(A and B) and P(neither).
Show answers (try them yourself first)
- 15/6
- 20.7
- 30.55
- 44/13
- 50.3
- 62/3
- 711/26
- 80.2 and 0.3
Rules of probability
A weather forecast of 130% chance of rain would sound wrong, and rightly so. Probabilities follow simple rules.
Rules: (1) 0 ≤ P(E) ≤ 1; (2) P(S) = 1; (3) P(∅) = 0; (4) for mutually exclusive events, P(A ∪ B) = P(A) + P(B). All outcome probabilities of a sample space add up to 1.
Formula: Σ P(outcomes) = 1
Rule 1: Missing probability — Total must be 1.
- Easy: Probabilities 0.1, 0.2, 0.3 and x: x = 0.4.
- Medium: A loaded die with P(1) = P(2) = P(3) = 0.1, P(4) = P(5) = 0.2: P(6) = 0.3; P(even) = 0.1 + 0.2 + 0.3 = 0.6.
- Exam-type: P(A) = 1/3, P(B) = 1/4 (exclusive): P(A or B) = 7/12; P(neither) = 5/12.
Rule 2: Is it valid? — Check each rule.
- Easy: 0.3, 0.4, 0.5 for 3 outcomes: total 1.2, not valid.
- Medium: A value like 1.2 or −0.1 cannot be a probability.
- Exam-type: 0.2, 0.3, 0.5: total 1, valid.
Check that the probabilities of all outcomes add to exactly 1.
Writing a probability as a percentage above 100% or below 0.
All parts make one whole.
Practice questions
- 1What is P(S)?
- 2What is the probability of an impossible event?
- 3Can a probability be 1.2?
- 4Four exclusive and exhaustive outcomes have probabilities 0.1, 0.2, 0.3 and x. Find x.
- 5What do the probabilities of all outcomes of a sample space add up to?
- 6A loaded die: P(1) = P(2) = P(3) = 0.1, P(4) = P(5) = 0.2 and P(6) = x. Find x and P(even).
- 7A and B are mutually exclusive, P(A) = 1/3, P(B) = 1/4. Find P(A or B) and P(neither).
- 8Can 0.3, 0.4 and 0.5 be the probabilities of the three outcomes of a sample space?
Show answers (try them yourself first)
- 11
- 20
- 3No
- 40.4
- 51
- 60.3 and 0.6
- 77/12 and 5/12
- 8No, they add to 1.2