Meaning of Probability

Before you roll a die you cannot know which face turns up, yet the chance of each one can be pinned down as an exact number. Probability expresses how likely something is as a value between 0 and 1, and when every outcome is equally likely it is simply the share of favorable cases among all of them. How probabilities add or overlap when you combine events follows naturally from this same idea. Drag the slider to grow the number of elements in event A and watch P(A) rise, then switch the Venn diagram between union, intersection, and complement.

Intuition of Probability

🎲 What is Probability?

①Likelihood of a certain outcome in an experiment

②Ratio of desired outcomes among all possible outcomes

③0 ≤ P(A) ≤ 1; certain → 1, impossible → 0

Sample Space & Events

Elements in event A3

Classical Probability

P(A) = n(A)n(S) = |A||S|

When all elementary outcomes are equally likely

Event Operations

Venn diagram0union

Addition Rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Probability of union (subtract overlap once)

Complement

P(Aᶜ) = 1 − P(A)

Probability that A does not occur

Mutually Exclusive vs Independent

Mutually Exclusive

A ∩ B = ∅ → P(A ∪ B) = P(A) + P(B)

Cannot happen together (empty intersection)

Independent

P(A ∩ B) = P(A) × P(B)

Each event does not affect the other

⚠️ Exclusive ≠ Independent

①Exclusive: cannot occur together (A∩B = ∅)

②Independent: no influence (P(A∩B) = P(A)P(B))

③Mutually exclusive → strongly dependent

④Confusing the two costs marks!

Work It Out

Example 1

Find the probability of rolling a multiple of 3 on one die.

Classical probability is (favorable outcomes)/(total). Multiples of 3 are 3 and 6, so 2 outcomes.

P = 2/6

Reduce.

= 1/3

▸ 1/3

Classical probability = favorable outcomes ÷ total outcomes.

Example 2

Choosing one integer from 1 to 10, find the probability it is a multiple of 2 or of 3.

Use the addition rule. There are 5 multiples of 2, 3 of 3, and 1 of 6.

P(A∪B) = P(A) + P(B) − P(A∩B)

Substitute.

= 5/10 + 3/10 − 1/10 = 7/10

▸ 7/10

For a union, the addition rule subtracts the intersection once.

Wrap-up

Core

P(A) = n(A)n(S), P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Classical probability and addition rule

Grade-11 school exam type

Tossing 3 coins at once, what is the probability of at least one head?

①1/2

②5/8

③3/4

④7/8

⑤1

▸ ④ 7/8

"At least one" is handled via the complement (all tails).

P(at least one head) = 1 − P(all tails)

The probability of all tails is (1/2)³.

= 1 − (1/2)³ = 1 − 1/8 = 7/8

🎯 Exam Points

①P(A) = n(A)/n(S) — classical probability

②Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B)

③Complement: P(Aᶜ) = 1 − P(A) (helpful for "at least")

④Distinguish exclusive vs independent

⑤Independent: P(A∩B) = P(A)P(B)

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Binomial Theorem

Conditional Probability

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