EQN-05-03

Probability Basics

Probability as favourable-over-total, the complement shortcut, and how to combine simple events.

~5 min read•Difficulty 2/5

P (event) = favourable outcomes / total outcomes

Every probability is between 0 (impossible) and 1 (certain). A fair die shows a "4" with probability 1/6.

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

"At least one" problems are usually fastest through the complement: P(at least one $) = 1 - P ($ none).

AND (both happen, independent): multiply. P(A and $B) = P (A) \times P (B) .$
OR (mutually exclusive): add. P(A or $B) = P (A) + P (B) .$

Two coin flips, both heads : \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} .

Rolling a 2 OR a 5 on a die : 1/6 + 1/6 = 2/6 = 1/3.

If an item is not replaced, the total drops for the next draw — recompute the denominator.

Diagram

P(red $) =$ favourable $\div$ total $= 4 \div 10.$

Worked examples

Example 1 (difficulty 1/5)

A bag has 4 red and 6 blue marbles. What is the probability of drawing a red marble?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A fair coin is flipped twice. What is the probability of at least one head?

◎ Predict the answer first:

Adding probabilities for "AND" events

Why: Combining feels like adding.

Fix: Independent "both happen" probabilities multiply.

Forgetting to drop the total when drawing without replacement

Why: The first denominator is reused.

Fix: Each removed item shrinks the pool for the next draw.