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.

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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.

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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.

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Equally Likely Outcomes

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

$P(A)$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
• $P(B)$P(B) = 2/11

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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

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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.

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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

Worked Example 7

A bag contains 12 counters. 4 are red, 3 are green and the rest are yellow. One counter is picked at random. Find P(yellow) as a decimal.

Solution:

• Yellow counters = 12 − 4 − 3 = 5
• P(yellow) = 5/12
• As a decimal: 5 ÷ 12 = 0.416… (or 0.42 to 2 d.p.)

Worked Example 8

Convert each probability to the specified form: (a) 7/20 to a percentage (b) 0.08 to a fraction in its simplest form (c) 64% to a decimal

Solution:

(a) 7/20 = 7 ÷ 20 = 0.35 = 35%

(b) 0.08 = 8/100 = 2/25

(c) 64% = 64/100 = 0.64

Worked Example 9

A weather forecaster says "there is a 1 in 5 chance of snow tomorrow". Write this probability as: (a) a fraction (b) a decimal (c) a percentage

Then state the probability it will not snow tomorrow as a percentage.

Solution:

(a) "1 in 5" = 1/5

(b) 1/5 = 1 ÷ 5 = 0.2

(c) 0.2 × 100 = 20%

P(not snow) = 1 − 0.2 = 0.8 = 80%

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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 $P(A)$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.

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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)

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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: $P(A)$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.

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Extended Worked Examples

Worked Example 4 — Listing outcomes with a sample space table

A fair coin is tossed and a fair four-sided dice (numbered 1–4) is rolled. Find the probability of obtaining a head and an even number.

Solution:

Set up a sample space table listing every equally likely outcome:

	1	2	3	4
H	H1	H2	H3	H4
T	T1	T2	T3	T4

There are $2 \times 4 = 8$2×4=8 equally likely outcomes. The favourable outcomes (head and even number) are H2 and H4, giving 2 outcomes.

$P(\text{head and even}) = \frac{2}{8} = \frac{1}{4}$P(head and even)=82​=41​

The theoretical probability is $\frac{1}{4}$41​ because we are assuming the coin and dice are fair.

Worked Example 5 — Probability from a Venn diagram

In a class of 30 students, 18 play football (F) and 12 play cricket (C). 7 play both sports. A student is picked at random. Find the probability that the student plays neither sport.

Solution:

Draw a Venn diagram with two overlapping circles F and C inside a universal set rectangle of 30 students.

• Intersection $F \cap C$F∩C = 7
• Football only = $18 - 7 = 11$18−7=11
• Cricket only = $12 - 7 = 5$12−7=5
• Total in at least one set = $11 + 7 + 5 = 23$11+7+5=23
• Neither = $30 - 23 = 7$30−23=7

$P(\text{neither}) = \frac{7}{30}$P(neither)=307​

Worked Example 6 — Probability scale reasoning

A weather app says the probability of rain tomorrow is 85%. A student says "so it is certain to rain". Explain why the student is wrong.

Solution: A probability of 1 (or 100%) means certain. 85% = 0.85, which is less than 1, so rain is very likely but not certain. There is still a probability of $1 - 0.85 = 0.15$1−0.85=0.15 that it will not rain.

Edexcel Exam Tip: Commentary questions like this are worth 1 or 2 marks on Foundation papers. Use the words impossible, unlikely, even chance, likely, certain precisely — and never say "certain" unless the probability is exactly 1.

Answering at different grade levels

Exam-style question: A bag contains 5 red, 3 blue and 2 green counters. A counter is picked at random. (a) Write down the probability that the counter is red. (b) Explain why the event "the counter is yellow" has probability 0. (c) The counter is replaced. A second counter is picked. Are these two events independent? Justify your answer.

Grades 3–4 answer:

(a) Total = $5 + 3 + 2 = 10$5+3+2=10. P(red) = $\frac{5}{10} = \frac{1}{2}$105​=21​. (b) There are no yellow counters in the bag, so it is impossible. Probability = 0. (c) Yes, because we put the first counter back.

Grades 5–6 answer:

(a) The sample space has 10 equally likely outcomes. P(red) = $\frac{5}{10} = \frac{1}{2}$105​=21​ (using theoretical probability). (b) Yellow is an impossible event within this sample space — no outcome satisfies the event, so the number of favourable outcomes is 0, giving probability $\frac{0}{10} = 0$100​=0. (c) The events are independent because replacing the counter restores the bag to its original state, so the probability of red on the second pick is unaffected by the first outcome.

Grades 7–9 answer:

(a) Using theoretical probability with equally likely outcomes: $P(R) = \frac{5}{10} = \frac{1}{2}$P(R)=105​=21​. (b) The event "yellow" is not in the sample space $\{R, B, G\}${R,B,G}; it is an empty event, so $P(\text{yellow}) = 0$P(yellow)=0. This is the lower bound of the probability scale. (c) Because the counter is replaced, the two draws are independent events: $P(R_2 \mid R_1) = P(R_2) = \frac{1}{2}$P(R2​∣R1​)=P(R2​)=21​. If the counter were not replaced, the events would cease to be independent and the second probability would become conditional probability [H] on the first outcome.

Edexcel alignment: This content is aligned with Edexcel GCSE Mathematics (1MA1) specification — specifically Topic P (P1 Probability language, P3 Sample space, P6 Mutually exclusive/independent). Assessed on Papers 1, 2, 3.