Probability

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 14 Probability

Lesson objectives

By the end of this topic, you should be able to:

• Define probability and determine it from experiments and real situations.
• Construct a probability space and use it to find probabilities.
• Apply the addition and multiplication rules.
• Use tree diagrams to solve problems, including without replacement.

Probability

Probability measures how likely an event is, on a scale from 0 to 1.

a) Theoretical and experimental

Theoretical probability = favourable outcomes ÷ total equally likely outcomes.

Experimental probability = successes ÷ trials.

The experimental value approaches the theoretical one as trials increase.

Some situations have no theoretical value at all. A drawing pin has no symmetry to reason from, so only experiment gives an answer.

b) The probability space

The probability space or sample space lists every possible outcome.

For two dice, a 6 × 6 grid gives all 36 outcomes. Counting the favourable ones gives any probability directly.

All probabilities in the space sum to 1.

c) Complementary events

P(not A) = 1 − P(A).

When "at least one" appears in a question, the complement is usually far quicker. The opposite of "at least one" is "none".

d) The addition rule

For mutually exclusive events, which cannot both happen: P(A or B) = P(A) + P(B).

Otherwise subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).

Drawing a red card or a king needs the subtraction, since the red kings would be counted twice.

e) The multiplication rule

For independent events, where one does not affect the other: P(A and B) = P(A) × P(B).

For dependent events: P(A and B) = P(A) × P(B given A).

Tossing a coin twice is independent. Drawing two cards without replacement is dependent.

f) Tree diagrams

Each branch carries its probability.

Probabilities on branches from the same point sum to 1.

Multiply along a path to get the probability of that whole sequence.

Add down the paths that satisfy the event.

g) With and without replacement

With replacement, the item goes back, so the second probability equals the first. The events are independent.

Without replacement, both the favourable count and the total drop by one. The events are dependent.

Take a bag of 5 red and 3 blue. Drawing two reds without replacement gives 5/8 × 4/7 = 20/56, which simplifies to 5/14.

With replacement it would be 5/8 × 5/8 = 25/64.

Read the question carefully. That single word changes every branch after the first.

h) Where this is used

Insurance and risk. Genetics. Quality control. Weather forecasting. Any decision made under uncertainty.

Words to know

• Probability -- a measure of how likely an event is, from 0 to 1.
• Probability space -- the set of all possible outcomes.
• Mutually exclusive -- events that cannot both occur.
• Independent events -- events where one does not affect the other's probability.
• Complement -- the event that a given event does not happen.

:::checkpoint Check yourself

1. What do all the probabilities in a sample space sum to?
2. A bag has 4 white and 6 black balls. Find P(both white) with replacement.
3. Find P(both white) without replacement.
4. When must you subtract an overlap in the addition rule? :::

Bridge to practice

The exercises begin with the probability scale and simple calculations, move through probability spaces, complements and the two rules, and finish with tree diagrams and selection without replacement. For every two-stage question, state first whether the item is replaced.