Counting & Probability

Probability measures how likely something is, from 0 to 1. Rolling a die gives six outcomes, so the favorable ones over the total is the probability. You add separate cases and multiply cases in a row, and a coin-toss tree shows why outcomes double each toss. Slide the die to watch the probability grow, or grow a tree to see outcomes multiply.

Intuition — All Possible Outcomes

At Most (probability of ≤ n)3

👀 What is Probability?

①Rolling a die gives 1–6 (six outcomes)

②"≤ 3" → 1, 2, 3 = three outcomes

③Probability = favorable outcomes ÷ total outcomes = 3/6

Counting — Sum Rule & Product Rule

Number of Tosses2

Sum Rule

Two events that cannot occur together → add the counts

A OR B → n(A) + n(B)

Product Rule

Two events occurring in sequence → multiply the counts

A AND B → n(A) × n(B)

Basic Properties of Probability

Definition

P(A) = n(A: favorable outcomes)n(S: all outcomes)

0 ≤ P(A) ≤ 1

💡 Properties

①Certain event: P = 1

②Impossible event: P = 0

③All probabilities lie between 0 and 1

④P(A) + P(not A) = 1

Computing Probabilities

Complement Probability

P(A^c) = 1 - P(A)

P(not A) = 1 - P(A)

📝 Using the Complement

①"P(at least one heads)" can be hard to count

②Use the complement instead

③P(at least one) = 1 - P(all tails) = 1 - 1/4 = 3/4

Work It Out

Example 1

Rolling one die, how many outcomes are even or a multiple of 3?

Even is {2, 4, 6} (3), multiples of 3 are {3, 6} (2), and the overlap (6) is 1.

3 + 2 − 1

Subtract the overlap once.

= 4 outcomes

▸ 4 outcomes

For "A or B", use the addition rule but subtract the overlap once.

Example 2

Rolling one die, find the probability of a prime number.

Primes are {2, 3, 5} (3 of them) out of 6 total.

P = 3/6

Reduce.

= 1/2

▸ 1/2

Probability = (favorable outcomes) ÷ (total outcomes).

Exam Wrap-up

Core Formulas

P(A) = n(A)n(S), P(A^c) = 1 - P(A)

Counting outcomes correctly is the start of every probability problem

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

Use the multiplication rule: the coin has 2 outcomes, the die 6.

2 × 6

Compute.

= 12 outcomes

🎯 Exam Points

①Sum vs Product rule: "or" → sum, "and" → product

②Draw a tree to count systematically

③n coin tosses → 2^n outcomes; n dice → 6^n

④Use complement when "at least one"

⑤Probability is always 0–1; if > 1 something is wrong

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