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Mathematics — DATA HANDLING & PROBABILITY

Subtopic: Probability (Age 13, Kenya)

Learning goals:

Understand basic probability words and notation.
Find probabilities when outcomes are equally likely.
Use simple examples: coin, dice, coloured balls, spinner.
Use experimental (relative frequency) and theoretical probability.

Key terms

Experiment: an action that gives outcomes (e.g., tossing a coin).
Outcome: a single result (e.g., Head).
Sample space (S): all possible outcomes (e.g., S = {H, T}).
Event: any collection of outcomes (e.g., getting an even number when rolling a die).
Favourable outcomes: outcomes in the event we want.
Theoretical probability: probability found by reasoning (when outcomes equally likely).
Experimental probability: probability found by doing many trials; also called relative frequency.

Basic formula

If all outcomes in S are equally likely, then:

P(event) = (number of favourable outcomes) ÷ (total number of outcomes in S)

Range: 0 ≤ P(event) ≤ 1. 0 means impossible, 1 means certain.

Probability scale (visual):

0.5

The shaded part shows an example probability of 0.5 (like tossing a fair coin and getting heads).

Simple examples

1) Tossing a fair coin

Sample space S = {H, T}. Number of outcomes = 2. Favourable outcomes for "Head" = 1.

P(Head) = 1 ÷ 2 = 1/2 = 0.5

H T

2) Rolling a fair 6-sided die

S = {1,2,3,4,5,6}. Total outcomes = 6.

Find P(rolling an even number). Favourable = {2,4,6} → 3 outcomes.

P(even) = 3 ÷ 6 = 1/2 = 0.5

3) Drawing a ball from a bag

A bag has 5 red and 3 blue balls. Total = 8. Find P(red).

P(red) = 5 ÷ 8

Complement rule

If A is an event, the complement A' is "A does not happen".

P(A') = 1 − P(A)

Example: P(getting a 6 when rolling a die) = 1/6, so P(not 6) = 1 − 1/6 = 5/6.

Experimental (relative frequency) probability

Do the experiment many times and record results. The experimental probability ≈ (number of times event happened) ÷ (total trials).

Example: Toss a coin 100 times and get 56 heads → experimental P(head) = 56/100 = 0.56. With more trials it should get closer to theoretical 0.5 for a fair coin.

Mutually exclusive and independent (short)

Mutually exclusive: cannot happen together (e.g., getting 2 and 3 on one die roll). P(A or B) = P(A) + P(B).
Independent: one event does not affect the other (e.g., toss coin then roll die). P(A and B) = P(A) × P(B).

Worked problem 1

A fair spinner has 4 equal sections coloured red, green, yellow and blue. What is P(spinning green)?

Solution: Total outcomes = 4, favourable = 1 (green). P(green) = 1 ÷ 4 = 1/4 = 0.25.

Worked problem 2

A bag contains 2 white, 3 black and 5 brown marbles (total 10). A marble is drawn at random.

Find P(black or brown).

Favourable outcomes = black (3) + brown (5) = 8. Total = 10.

P(black or brown) = 8 ÷ 10 = 4/5 = 0.8.

Practice questions

Toss a fair coin twice. List the sample space and find P(getting exactly one Head).
Roll a die. Find P(number less than 3).
A box has 4 green and 6 white sweets. One sweet is picked at random. Find P(white).
A spinner has 3 equal red, 1 yellow, 2 blue sectors. Find P(red).

Answers (click to view)

Sample space = {HH, HT, TH, TT}. Exactly one Head = {HT, TH} → P = 2/4 = 1/2.
Numbers less than 3: {1,2} → P = 2/6 = 1/3.
P(white) = 6/10 = 3/5 = 0.6.
Total sectors = 3 + 1 + 2 = 6. P(red) = 3/6 = 1/2.

Tips for learners

Always write the sample space first.
Count favourable outcomes carefully.
Simplify fractions where possible.
Use experiments (many trials) to check theoretical answers.

Good luck! Practice with coins, dice, spinners and coloured counters to get confident with probability.

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