Subject: Mathematics | Level: JHS (Basic 7 – 9) | Topic: Probability | Difficulty: Intermediate
Introduction
Probability is the Mathematics of chance. It measures how likely an event is to happen. Every time you toss a coin, roll a die, pick a card from a pack, or check the weather forecast, probability is at work. In the BECE, probability questions are generally straightforward once you understand the key definitions and formulas. This lesson covers everything you need to know about probability at the JHS level.
1. Key Definitions
Experiment: Any action that produces a result. Example: tossing a coin, rolling a die.
Outcome: A single possible result of an experiment. Example: getting a Head when tossing a coin.
Sample space (S): The set of all possible outcomes of an experiment.
Example: Tossing a coin → S = {Head, Tail}
Example: Rolling a die → S = {1, 2, 3, 4, 5, 6}
Event: A specific outcome or set of outcomes we are interested in.
Example: Getting an even number when rolling a die → Event = {2, 4, 6}
2. The Probability Formula
P(Event) = Number of favourable outcomes ÷ Total number of possible outcomes
Probability is always a value between 0 and 1 (inclusive):
- P = 0 means the event is impossible
- P = 1 means the event is certain
- P = ½ (0.5) means the event is equally likely to happen or not happen
Probability can also be expressed as a fraction, decimal, or percentage.
3. Worked Examples
Example 1: A fair coin is tossed once. What is the probability of getting a Head?
Total outcomes = 2 (Head or Tail)
Favourable outcomes = 1 (Head)
P(Head) = 1/2 or 0.5 or 50%
Example 2: A die is rolled once. What is the probability of getting a 4?
Total outcomes = 6 (1, 2, 3, 4, 5, 6)
Favourable outcomes = 1 (the number 4)
P(4) = 1/6
Example 3: A die is rolled. Find the probability of getting an even number.
Even numbers on a die: {2, 4, 6} → 3 favourable outcomes
P(even) = 3/6 = 1/2
Example 4: A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random. Find:
(a) P(red ball) = 5/10 = 1/2
(b) P(blue ball) = 3/10
(c) P(green ball) = 2/10 = 1/5
(d) P(not red) = 1 – 1/2 = 1/2
Check: 1/2 + 3/10 + 1/5 = 5/10 + 3/10 + 2/10 = 10/10 = 1 ✓ (probabilities of all outcomes must sum to 1)
4. Complementary Events
The complement of an event A is “not A” – all outcomes where A does NOT occur.
P(not A) = 1 – P(A)
Example: The probability that it will rain tomorrow is 0.35. What is the probability it will NOT rain?
P(not rain) = 1 – 0.35 = 0.65
5. Mutually Exclusive Events
Two events are mutually exclusive if they cannot happen at the same time.
Example: Getting a Head AND a Tail on a single coin toss – impossible, so they are mutually exclusive.
P(A or B) = P(A) + P(B) (for mutually exclusive events)
Example: A die is rolled. Find the probability of getting a 2 or a 5.
P(2) = 1/6; P(5) = 1/6
P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
6. Independent Events
Two events are independent if the outcome of one does not affect the other.
Example: Tossing a coin and rolling a die – the result of the coin does not affect the die result.
P(A and B) = P(A) × P(B) (for independent events)
Example: A coin is tossed and a die is rolled. Find P(Head and 6).
P(Head) = 1/2; P(6) = 1/6
P(Head and 6) = 1/2 × 1/6 = 1/12
7. Experimental vs Theoretical Probability
Theoretical probability is calculated using the formula above – it predicts what should happen.
Experimental (relative frequency) probability is based on actual results from carrying out an experiment.
Experimental probability = Number of times event occurred ÷ Total number of trials
Example: A coin was tossed 200 times and Heads appeared 95 times.
Experimental P(Head) = 95/200 = 19/40
Theoretical P(Head) = 1/2 = 20/40
The more times an experiment is repeated, the closer experimental probability gets to theoretical probability.
Summary
- P(Event) = Favourable outcomes ÷ Total outcomes
- Probability is always between 0 (impossible) and 1 (certain)
- P(not A) = 1 – P(A)
- Mutually exclusive events: P(A or B) = P(A) + P(B)
- Independent events: P(A and B) = P(A) × P(B)
- All probabilities in a sample space must sum to 1
Practice Questions
- A bag contains 4 yellow beads, 6 red beads, and 2 white beads. A bead is drawn at random. Find P(yellow), P(red), and P(not white).
- A fair die is rolled. Find P(prime number).
- The probability that a student passes Mathematics is 0.7 and the probability of passing English is 0.8. Find the probability that the student passes both subjects (assuming they are independent).
- A card is drawn from the numbers 1-10. Find P(multiple of 3).
- In 150 throws of a die, the number 4 came up 30 times. What is the experimental probability of getting a 4? How does this compare to the theoretical probability?
Answers: (1) P(yellow)=4/12=1/3; P(red)=6/12=1/2; P(not white)=10/12=5/6 (2) Primes on a die: 2,3,5 → P=3/6=1/2 (3) 0.7×0.8=0.56 (4) Multiples of 3 between 1-10: 3,6,9 → P=3/10 (5) Experimental=30/150=1/5=0.2; Theoretical=1/6≈0.167; Experimental is close but slightly higher this time
Previous Lesson: JHS Mathematics – Statistics: Mean, Median, Mode, and Data Handling
Next Lesson: JHS Mathematics – Vectors and Translation