Learning Resources

Probability

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 08 Pathway: N/A

First Principles

Objective: Find the theoretical and experimental probability of simple events and express answers accurately as fractions in simplest form, decimals, or percentages.

Concrete scenario: Imagine spinning a colourful prize wheel at a trade fair in Nairobi. The wheel is divided into 4 equal sectors: Red, Blue, Green, and Yellow. Because each sector is identical in size, every colour is equally likely to be selected. When you spin the wheel once, the chance of landing on Red is 1 out of 4, written as \(\frac{1}{4}\), \(0.25\), or \(25\%\).

Geometric insight: Probability measures the share of opportunity. On the spinner, Red occupies \(\frac{1}{4}\) (or \(90^\circ\)) of the full \(360^\circ\) circle. When all outcomes are equal slices of the sample space, probability becomes a simple ratio of counts.

Algebraic definition: For any event \(E\) with equally likely outcomes in a sample space \(S\):

\[ P(E) = \frac{n(E)}{n(S)} = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}} \]

The probability of any event is always bounded between \(0\) (impossible) and \(1\) (certain):

\[ 0 \le P(E) \le 1 \]

Interactive Spinner: Law of Large Numbers

Spin the wheel and watch how experimental probability approaches the theoretical value of \(25\%\) (\(\frac{1}{4}\)) as the number of trials increases.

Colour	Count	Fraction	%

Total Spins: 0

Key Formulas

\[ P(E) = \frac{n(E)}{n(S)} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} \]

This classical formula applies when all elementary outcomes in the sample space \(S\) are equally likely.

\[ P(E') = 1 - P(E) \]

Complement Rule: The probability that event \(E\) does not occur is equal to \(1\) minus the probability that \(E\) occurs.

\[ P(A \cup B) = P(A) + P(B) \quad \text{(if } A \text{ and } B \text{ are mutually exclusive)} \]

Addition Rule: If two events cannot happen at the same time (disjoint), the probability of either occurring is the sum of their individual probabilities.

\[ P(A \cap B) = P(A) \times P(B) \quad \text{(if } A \text{ and } B \text{ are independent)} \]

Multiplication Rule: For independent successive events where the outcome of the first does not change the likelihood of the second.

\[ \text{Decimal} = \frac{\text{Numerator}}{\text{Denominator}}, \quad \text{Percentage} = \text{Decimal} \times 100\% \]

Convert seamlessly between forms: \(\frac{3}{5} = 0.60 = 60\%\).

Worked Examples

Worked Example 1 (Easy — Single Die Roll):
A fair 6-sided die is rolled once during a board game in Nakuru. Find the probability of rolling a prime number. Express your answer as a fraction in simplest form, a decimal, and a percentage.

Identify the sample space \(S\): \(S = \{1, 2, 3, 4, 5, 6\}\), so \(n(S) = 6\).
Identify favourable outcomes \(E\): The prime numbers on a die are \(\{2, 3, 5\}\), so \(n(E) = 3\).
Apply the formula: \[ P(\text{Prime}) = \frac{n(E)}{n(S)} = \frac{3}{6} = \frac{1}{2} \]
Convert to decimal and percentage: \(\frac{1}{2} = 0.5 = 50\%\).
Final Answer: \(\frac{1}{2}\), \(0.5\), or \(50\%\).

Worked Example 2 (Medium — Complement & Selection):
A carton at a local supermarket in Eldoret contains 8 mango juices, 5 passion juices, and 7 guava juices. If a customer picks one juice can at random, what is the probability that it is neither mango nor guava? Express your answer as a fraction and percentage.

Calculate total items \(n(S)\): \(n(S) = 8 + 5 + 7 = 20\) juice cans.
Identify favourable outcomes: Neither mango nor guava means the juice must be passion. \(n(\text{Passion}) = 5\).
Calculate probability: \[ P(\text{Passion}) = \frac{5}{20} = \frac{1}{4} \]
Verify using complement rule: \[ P(\text{Mango or Guava}) = \frac{8 + 7}{20} = \frac{15}{20} = \frac{3}{4} \] \[ P(\text{Neither}) = 1 - \frac{3}{4} = \frac{1}{4} = 25\% \]
Final Answer: \(\frac{1}{4}\) or \(25\%\).

Worked Example 3 (Hard — Two-Stage Selection Without Replacement):
A beadwork artisan in Machakos has a bag containing 4 red beads and 6 green beads. She draws two beads consecutively without replacement. What is the probability that she draws at least one red bead?

Find total beads: \(4 + 6 = 10\) beads.
Use the complement strategy: "At least one red" is the opposite of "both beads are green". \[ P(\text{at least 1 red}) = 1 - P(\text{both green}) \]
Calculate probability of first green bead: \(P(G_1) = \frac{6}{10}\).
Calculate probability of second green bead (without replacement): After 1 green bead is removed, 5 green beads remain out of 9 total beads: \(P(G_2 \mid G_1) = \frac{5}{9}\).
Multiply probabilities: \[ P(\text{both green}) = \frac{6}{10} \times \frac{5}{9} = \frac{30}{90} = \frac{1}{3} \]
Subtract from 1: \[ P(\text{at least 1 red}) = 1 - \frac{1}{3} = \frac{2}{3} \]
Final Answer: \(\frac{2}{3}\) (or approximately \(66.7\%\)).

Common Mistakes

Mistake Assuming that because there are only two outcomes, each has a 50% probability (e.g., "It will either rain tomorrow in Nairobi or it will not, so the probability of rain is 50%").

Why it feels right People associate binary choices with coin flips, where outcomes are symmetrical.

Correction Outcomes must be proven equally likely before dividing by the count. The probability of rain depends on humidity, pressure, and seasonal climate — not merely the number of categories.

Mistake Adding probabilities for overlapping (non-mutually exclusive) events without subtracting the intersection, leading to values greater than 1.

Why it feels right Beginners directly apply the addition rule \(P(A) + P(B)\) without checking if an element belongs to both sets (e.g., picking a card that is both red and a King).

Correction Always ensure \(0 \le P(E) \le 1\). For overlapping events, subtract the common outcomes: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

Mistake Forgetting to reduce the sample space when sampling without replacement.

Why it feels right It is easy to keep using the initial total denominator for every consecutive pick.

Correction If an item is kept after being drawn, both the favourable count (if matching) and the total count decrease by 1 for subsequent draws.

Real World

The Inspection Process: In a crate of 50 frozen Nile perch fillets, 5 fillets are sampled at random. If historical batch records show a standard defective rate of \(2\%\), probability models help inspectors calculate the risk of accepting a sub-standard shipment.

Weather Forecasting in Agriculture: Kenya Meteorological Department (KMD) provides farmers in Trans-Nzoia with precipitation probabilities (e.g., "\(70\%\) chance of rain"). A \(70\%\) chance means that under identical atmospheric measurements historically recorded, rainfall occurred on 7 out of 10 days. Farmers use this to schedule maize planting and fertilisation.

Telecommunications & M-Pesa Security: One-Time PINs (OTPs) utilize sample spaces of digits (0–9). A 4-digit numeric code has \(10^4 = 10,000\) equally likely combinations, ensuring the probability of a random guess succeeding on the first attempt is tiny: \(\frac{1}{10,000} = 0.01\%\).

Practice

A basket at Kongowea Market contains 5 ripe mangoes, 3 green mangoes, and 2 avocados. If Achieng picks one fruit at random, what is the probability that she picks a green mango? (Type only the fraction in simplest form, e.g., 3/10)

Review the concepts above.

A fair standard 6-sided die is rolled once during a game. What is the probability of rolling a number strictly greater than 4? (Type only the fraction in simplest form, e.g., 1/3)

Review the concepts above.

A SACCO fleet in Kisumu has 25 matatus. Records show that 15 are white, 6 are blue, and 4 are yellow. If a matatu is selected at random, what is the probability that it is NOT white? Express your answer as a percentage without the % sign. (Type only the number, e.g., 40)

Review the concepts above.

A fair 10-shilling coin is tossed 3 times. What is the probability of getting exactly two heads? (Type only the fraction in simplest form, e.g., 3/8)

Review the concepts above.

A box contains 4 red pens and 6 blue pens. Juma picks one pen at random, keeps it, and then picks a second pen. What is the probability that BOTH pens picked are blue? (Type only the fraction in simplest form, e.g., 1/3)

Review the concepts above.

A pouch contains 4 red beads and 6 green beads. If two beads are drawn one after another without replacement, what is the probability that AT LEAST ONE bead drawn is red? (Type only the fraction in simplest form, e.g., 2/3)

Review the concepts above.