Probability expresses how likely something is as a number from 0 to 1.
A die can land on any of the six faces from 1 to 6 with the same chance, so dividing the number of outcomes you want by 6 gives the probability.
The starting point of probability is therefore counting outcomes without missing or repeating any, and a tree diagram is the most dependable tool for that.
Every extra coin toss splits each branch of the tree in two, so the number of outcomes doubles each time; that is the product rule.
Every outcome that can happen
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The faces of a die that are n or less, and their probability
3
👀 What probability is
①A die shows one of the six faces from 1 to 6, and each face is equally likely.
②The faces 3 or less are 1, 2 and 3: three outcomes.
③Probability is the number of wanted outcomes divided by the number of all outcomes, so it is 3/6 = 1/2. Changing n in the picture changes the shaded dice and the length of the bar together.
Counting outcomes with a tree diagram
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A tree diagram that branches with each coin toss, with each final branch labelled by its result
2
Sum rule
two events A, B that cannot happen together → outcomes of A or B = m + n
If the two events can happen together, the overlapping outcomes must be subtracted once.
Product rule
A in m ways, and for each of them B in n ways → outcomes of A then B = m × n
In a tree diagram, the number of branches is multiplied at every stage.
Basic properties of probability
Meaning of probability
P(A) = n(A: favorable outcomes)n(S: all outcomes)
Used when all outcomes are equally likely; the value satisfies 0 ≤ P(A) ≤ 1.
💡 The range of probability
①An event that is certain has probability 1, like rolling a 6 or less on a die.
②An impossible event has probability 0, like rolling a 7.
③So the probability of any event is between 0 and 1 inclusive.
④The probability that A happens plus the probability that A does not happen is 1.
Calculating with the complement
Probability of the complement
P(Ac) = 1 - P(A)
probability that A does not happen = 1 − (probability that A happens)
📝 When you see "at least", use the complement
①Find the probability of at least one head when two coins are tossed.
②You could count HH, HT and TH one by one, but the opposite, "all tails", is only TT, which is far easier to count.
③P(at least one head) = 1 − P(all tails) = 1 − 1/4 = 3/4. The final branch written in red in the tree diagram is exactly this complement.
Worked examples
Example 1
Rolling one die, how many outcomes are even or a multiple of 3?
1
Even is {2, 4, 6} (3), multiples of 3 are {3, 6} (2), and the overlap (6) is 1.
3 + 2 − 1
2
Subtract the overlap once.
= 4 outcomes
▸ 4 outcomes
Overlapping events are counted with n(A)+n(B)−n(A∩B), not the exclusive addition rule.
Example 2
Rolling one die, find the probability of a prime number.
1
Primes are {2, 3, 5} (3 of them) out of 6 total.
P = 3/6
2
Reduce.
= 1/2
▸ 1/2
Probability = (favorable outcomes) ÷ (total outcomes).
Exam checklist
Key formula of probability
P(A) = n(A)n(S), P(Ac) = 1 - P(A)
Counting outcomes without missing or repeating any is where every probability problem begins.
Grade-8 school exam type
Tossing one coin and rolling one die together, how many total outcomes are there?
①6
②8
③12
④18
⑤36
▸ ③ 12
1
Use the multiplication rule: the coin has 2 outcomes, the die 6.
2 × 6
2
Compute.
= 12 outcomes
🎯 Exam Points
①The sum rule adds counts directly only when the two events cannot happen together; if they overlap, count n(A) + n(B) − n(A∩B).
②"One after another" or "at the same time" calls for the product rule.
③Tossing a coin n times gives 2ⁿ outcomes; rolling a die n times gives 6ⁿ.
④When you see "at least", count the opposite event and subtract from 1.
⑤A probability is always between 0 and 1, so a value greater than 1 means the counting went wrong.