Mathematics · Lesson 27

Probability in plain words

JAMBWAECNECO

What the syllabus asks

JAMB: Statistics: probability, simple and compound events, permutations and combinations (basic)

WAEC: Probability: experimental and theoretical probability, sample space, equally likely outcomes, mutually exclusive and independent events, addition and multiplication rules, complementary events, probability with and without replacement, tree diagrams and outcome tables

NECO: Probability: experimental and theoretical probability, sample space, equally likely outcomes, mutually exclusive and independent events, addition and multiplication rules, complementary events, probability with and without replacement, tree diagrams and outcome tables

What you will learn

Define probability and use the 0 to 1 scale.
List sample spaces using lists, tables and tree diagrams.
Calculate theoretical and experimental probabilities.
Use complementary, mutually exclusive and independent events correctly.
Solve problems with and without replacement.
Use outcome tables for two dice and Venn diagrams.
Find expected frequencies.

Key points

Probability is from 0 (impossible) to 1 (certain).
P(A) = favourable ÷ total, when outcomes are equally likely.
Relative frequency = successes ÷ trials; expected number = P × trials.
P(not A) = 1 − P(A); "at least one" = 1 − P(none).
Two dice: 36 outcomes; use a table.
Mutually exclusive: P(A or B) = P(A) + P(B); otherwise subtract P(A and B).
Independent: P(A and B) = P(A) × P(B).
Without replacement: reduce both the count and the total for later draws.
Tree diagrams: multiply along branches, add across paths; totals add to 1.
Permutations count arrangements (order matters); combinations count selections.

Start here: What is probability?

Every day we talk about chance. "It will likely rain in Port Harcourt this afternoon." "Our team has no chance of winning." Probability turns these feelings into numbers.

Probability is a number from 0 to 1 that measures how likely an event is to happen.
P = 0: impossible. P = 1: certain. P = ½: an even chance.

Probabilities can be written as fractions, decimals or percentages: ¼ = 0.25 = 25%. They can never be negative and never greater than 1. If you get P = 1.3 or P = −0.2 in an exam, you have made an error.

Key words

An experiment is any activity with uncertain results: tossing a coin, throwing a die, drawing a ball from a bag, picking a student at random.
An outcome is one possible result: "head", "5", "red ball".
The sample space is the set of all outcomes: for a die, S = {1, 2, 3, 4, 5, 6}.
An event is a collection of outcomes we are interested in: "an even number" = {2, 4, 6}.

Worked example 27.1

Place these on the probability scale: (a) the sun rising tomorrow (b) getting 7 on a normal die (c) a tossed coin showing head.

(a) certain: 1.
(b) impossible: 0 (no face shows 7).
(c) even chance: ½.

1, 0, ½

⚠️ Common mistake: writing probability as a ratio like "1 : 6". Write it as a fraction, 1/6.

Try it yourself. List the sample space when a coin is tossed twice.

Show solution

{HH, HT, TH, TT}.

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Practice questions

1. A die is thrown. Find P(a factor of 6).

Show worked answer

{1, 2, 3, 6}: 2/3.

2. Two coins are tossed. Find P(at least one tail).

Show worked answer

1 − ¼ = ¾.

3. Two dice are thrown. Find P(sum = 8).

Show worked answer

5/36: 5/36.

4. P(A) = 0.4, P(B) = 0.5, independent. Find P(A or B).

Show worked answer

0.4 + 0.5 − 0.2 = 0.7.

5. A bag has 6 blue and 4 red pens. Two are taken without replacement. Find P(both blue).

Show worked answer

6/10 × 5/9 = 1/3.

Key words

Experiment: An activity with uncertain results, such as tossing a coin.
Outcome: One possible result of an experiment.
Sample space: The set of all possible outcomes.
Event: A set of one or more outcomes.
Equally likely: Outcomes with the same chance of happening.
Probability: A number from 0 to 1 measuring how likely an event is.
Complementary events: An event and its "not"; their probabilities add to 1.
Mutually exclusive: Events that cannot happen together.
Independent events: Events where one does not affect the other.
With replacement: The item is put back before the next draw.
Without replacement: The item is not put back, so totals change.
Relative frequency: Number of times an event occurs ÷ number of trials.

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