Introduction to Probability

This lesson introduces the fundamental ideas of probability as required by the Edexcel GCSE Mathematics (1MA1) specification. You will learn how to describe the likelihood of events using the probability scale from 0 to 1, express probabilities as fractions, decimals and percentages, and understand the concepts of impossibility and certainty.

Probability appears across all three papers on the Edexcel 1MA1 exam — Paper 1 (non-calculator), Paper 2 (calculator) and Paper 3 (calculator). Foundation questions typically ask you to mark events on a probability scale or calculate simple probabilities, while Higher questions extend to algebraic probability and proof.

---

The Probability Scale

Probability measures how likely an event is to happen. Every probability lies on a scale from 0 to 1.

Probability	Meaning
0	The event is impossible — it cannot happen
Between 0 and 0.5	The event is unlikely
0.5	The event has an even chance — equally likely to happen or not
Between 0.5 and 1	The event is likely
1	The event is certain — it must happen

Key Point: A probability can never be negative and can never be greater than 1. If your answer falls outside the range 0 to 1, you have made an error.

---

Writing Probabilities as Fractions, Decimals and Percentages

Probabilities can be expressed in three equivalent forms:

Fraction	Decimal	Percentage	Likelihood
0	0	0%	Impossible
1/10	0.1	10%	Very unlikely
1/4	0.25	25%	Unlikely
1/2	0.5	50%	Even chance
3/4	0.75	75%	Likely
1	1.0	100%	Certain

Converting Between Forms

• Fraction to decimal: Divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375.
• Decimal to percentage: Multiply by 100. For example, 0.375 × 100 = 37.5%.
• Percentage to fraction: Write over 100 and simplify. For example, 37.5% = 375/1000 = 3/8.

Edexcel Exam Tip: On Paper 1 (non-calculator), you will need to convert between fractions, decimals and percentages without a calculator. Practise common conversions until they are automatic.

---

Equally Likely Outcomes

When all outcomes of an experiment are equally likely, the probability of an event A occurring is:

P(A) = number of favourable outcomes / total number of possible outcomes

This is sometimes called the theoretical probability.

Worked Example 1

A fair six-sided dice is rolled once. Find the probability of rolling a number greater than 4.

Solution:

• Total number of equally likely outcomes = 6 (the numbers 1, 2, 3, 4, 5, 6)
• Favourable outcomes (greater than 4) = {5, 6}, so there are 2 favourable outcomes
• P(greater than 4) = 2/6 = 1/3

Worked Example 2

A bag contains 3 red balls, 5 blue balls and 2 green balls. A ball is chosen at random. Find the probability that the ball is blue.

Solution:

• Total number of balls = 3 + 5 + 2 = 10
• Number of blue balls = 5
• P(blue) = 5/10 = 1/2

Worked Example 3

The letters of the word PROBABILITY are written on separate cards and placed in a bag. One card is drawn at random. Find the probability that the letter drawn is a B.

Solution:

• The word PROBABILITY has 11 letters: P, R, O, B, A, B, I, L, I, T, Y
• The letter B appears 2 times
• P(B) = 2/11

---

Key Vocabulary

Term	Definition
Event	A particular outcome or set of outcomes from an experiment
Outcome	A single possible result of an experiment
Equally likely	Outcomes that all have the same probability of occurring
Fair	A dice, spinner or coin where all outcomes are equally likely
Random	Every item has an equal chance of being selected
Trial	One performance of an experiment (e.g. one roll of a dice)
Experiment	A repeatable process that gives a set of outcomes

---

Impossibility and Certainty

• An impossible event has a probability of 0. For example, rolling a 7 on a standard six-sided dice.
• A certain event has a probability of 1. For example, rolling a number less than 7 on a standard six-sided dice.

Worked Example 4

A spinner has sections coloured red, blue, green and yellow. State the probability of spinning a purple.

Solution: There is no purple section on the spinner. The event is impossible, so P(purple) = 0.

Worked Example 5

A bag contains only red counters. One counter is drawn at random. What is the probability that it is red?

Solution: Every counter is red, so the event is certain. P(red) = 1.

---

Probabilities of All Outcomes Sum to 1

For any experiment, the probabilities of all possible outcomes must add up to 1.

If you know the probabilities of some outcomes, you can find the missing probability.

Worked Example 6

A biased spinner has four sections coloured red, blue, green and yellow. The table shows the probability of landing on each colour.

Colour	Red	Blue	Green	Yellow
Probability	0.3	0.25	0.15	?

Find the probability of landing on yellow.

Solution: P(yellow) = 1 − (0.3 + 0.25 + 0.15) = 1 − 0.7 = 0.3

---

Common Mistakes and Misconceptions

1. Probability greater than 1 or less than 0: This is impossible. Always check your answer lies between 0 and 1.
2. Confusing "likely" with "certain": An event with probability 0.9 is very likely but not certain.
3. Not simplifying fractions: While Edexcel will accept unsimplified fractions, simplifying shows strong mathematical communication and avoids errors in follow-up calculations.
4. Forgetting that "at random" means equally likely: When a question says "at random", you can use the formula P(A) = favourable/total.
5. Confusing probability with proportion: Probability is a number between 0 and 1; it is not the same as a frequency count.

---

Practice Problems

1. A fair coin is tossed. What is the probability of getting heads? (Answer: 1/2)

2. A bag contains 4 red, 6 blue and 5 yellow sweets. One sweet is picked at random. What is the probability it is yellow? (Answer: 5/15 = 1/3)

3. A fair six-sided dice is rolled. What is the probability of rolling an even number? (Answer: 3/6 = 1/2)

4. A spinner has sections numbered 1 to 8. What is the probability of spinning a prime number? (Answer: The primes are 2, 3, 5, 7 so P = 4/8 = 1/2)

5. The probability of event A is 0.35. Express this as a fraction in its simplest form. (Answer: 35/100 = 7/20)

6. A biased dice has P(1) = 0.1, P(2) = 0.15, P(3) = 0.2, P(4) = 0.2, P(5) = 0.15. Find P(6). (Answer: 1 − 0.8 = 0.2)

---

Summary

• Probability is measured on a scale from 0 (impossible) to 1 (certain).
• Probabilities can be written as fractions, decimals or percentages.
• For equally likely outcomes: P(A) = favourable outcomes / total outcomes.
• The probabilities of all possible outcomes sum to 1.
• Probability questions appear on all three Edexcel papers — always show your working clearly.