Introduction to Probability

The probability scale from 0 to 1, sample spaces, theoretical probability, coins, dice, and cards.

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Visual explanation and practical step-by-step walk-through

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Comprehensive Study Notes

Aligned with Ghana NaCCA & WAEC BECE syllabus standards

Topic Introduction & Real-World Context:

Probability is the mathematical study of chance, uncertainty, and likelihood. In real life, weather forecasts predict the chance of rain, doctors assess medical risks, and games of ludo or football rely on odds. In the BECE exam, questions on coins, fair 6-sided dice, and drawing colored marbles from a bag test your understanding of sample spaces and theoretical probability.

What You Will Master in This Lesson (NaCCA Objectives):

Define probability and describe the probability scale from 0 (impossible) to 1 (certain).

List the sample space for simple random experiments (coins, dice, colored cards).

Calculate theoretical probability: P(E) = n(E) / n(S).

Understand and apply complementary probability: P(not E) = 1 - P(E).

1. The Probability Scale

Probability measures how likely an event is to happen: • It is expressed as a proper fraction, decimal, or percentage between 0 and 1. • P = 0: Impossible event (e.g. rolling a 7 on a standard 6-sided die). • P = 1 (or 100%): Certain event (e.g. the sun rising in the east). • P = 0.5 (or ½): Even chance (e.g. getting Heads when tossing a fair coin). Rule: Probability can NEVER be negative, and can NEVER be greater than 1!

2. Sample Space and Theoretical Probability

• Sample Space (S): The set of ALL possible outcomes of an experiment. - Tossing a coin: S = {Heads, Tails}, n(S) = 2. - Rolling a die: S = {1, 2, 3, 4, 5, 6}, n(S) = 6. • Theoretical Probability Formula: P(Event E) = (Number of favorable outcomes) / (Total possible outcomes) P(E) = n(E) / n(S)

3. Complementary Events

The probability that an event will NOT happen is called its complement (E'): Formula: P(Not E) = 1 - P(E) Example: If the probability of passing a test is ⅘, the probability of failing is 1 - ⅘ = ⅕. Sum of all probabilities in a sample space always equals 1: P(E) + P(E') = 1.

Common Mistakes Students Make in BECE Examinations:

⚠️Giving an answer greater than 1 (e.g. 5/3) or negative. Probability must be between 0 and 1.

⚠️Leaving fractions unsimplified (e.g. leaving 2/6 instead of 1/3).

⚠️Misidentifying the total sample space n(S).

Teacher's BECE Exam Pro-Tips:

⭐Always count the total number of items first to find n(S).

⭐Always simplify your final probability fraction to lowest terms.

⭐Remember prime numbers on a die are {2, 3, 5} (1 is NOT prime!).

Quick Revision Summary Checklist:

Probability ranges from 0 (impossible) to 1 (certain).

P(E) = favorable outcomes / total outcomes.

P(not E) = 1 - P(E).

Always simplify fractions to lowest terms.

Step-by-Step Worked Examples (1)

Real BECE exam-standard problems with complete solution steps

Example 1: Probability with a 6-Sided Die

Problem StatementA fair 6-sided die is rolled once. Find the probability of getting: (i) an even number (ii) a prime number.

Step-by-Step Solution:

Step 1: Sample space S = {1, 2, 3, 4, 5, 6}, total n(S) = 6.

Step 2: Even numbers E = {2, 4, 6}, n(E) = 3. P(Even) = 3/6 = 1/2.

Step 3: Prime numbers P = {2, 3, 5}, n(P) = 3 (1 is not prime!). P(Prime) = 3/6 = 1/2.

💡

Key Takeaway / Exam Rule: Remember that 1 is neither prime nor composite.

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