Ch 1 · 8.1–8.2 Probability, Relative and Expected Frequency · IGCSE Mathematics 0580
Topic 1 of 3 in Probability — IGCSE Mathematics 0580 — 57 lessons.
Introduction
"There is a 30% chance of rain", "the dice seems to land on six too often": probability puts numbers on uncertainty. Sometimes you can calculate it from equally likely outcomes; sometimes you must estimate it from experiments. (Syllabus sections 8.1 and 8.2.)
Key ideas — The probability scale
Probabilities lie from 0 (impossible) to 1 (certain), written as fractions, decimals or percentages (not "1 in 4" or ratios). P(A) is the probability of event A.
🧑🏫 Teacher's explanation
Imagine a long ruler that starts at 0 and ends at 1. This is your probability scale. On the far left, at 0, an event is 'impossible'—like rolling a 7 on a standard six-sided die. On the far right, at 1, an event is 'certain'—like the sun rising tomorrow. Everything else, like flipping a coin or winning a game, falls somewhere in the middle.
📖 Definitions
🔍 Worked example — Easy
A bag contains 5 red marbles and 3 blue marbles. Find the probability, P(R), of picking a red marble.
🔍 Worked example — Standard
The probability that it rains tomorrow is 0.35. Calculate the probability that it does not rain tomorrow, P(R').
🔍 Worked example — Exam-style
A fair six-sided die is rolled 120 times. Calculate the expected frequency of rolling a 4.
⚠️ Common errors
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🎯 Exam question
(a) Find the probability of landing on Red. (b) Calculate the expected frequency of landing on Blue. [3 marks]
Key ideas — Single events
When outcomes are equally likely:
🧑🏫 Teacher's explanation
Probability is simply a way of measuring how likely something is to happen on a scale from 0 (impossible) to 1 (certain). Think of it like a thermometer for luck: 0 is freezing (it won't happen), 1 is boiling (it definitely will), and 0.5 is right in the middle, like a coin toss.
📖 Definitions
🔍 Worked example — Easy
A bag contains 5 red, 3 blue and 2 green counters. A counter is drawn at random. Calculate the probability that the counter is NOT blue.
🔍 Worked example — Standard
In an experiment, a spinner is spun 200 times. The number '4' comes up 35 times. Estimate the probability of spinning a '4' and use this to find the expected frequency of '4' in 500 spins.
⚠️ Common errors
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🎯 Exam question
Key ideas — The complement
P(A') = 1 - P(A)
🧑🏫 Teacher's explanation
Imagine you are looking at a pizza that has been cut into several slices. If you know that 3/4 of the pizza has been eaten, you don't need to count every single crumb to know how much is left. You simply subtract the eaten part from the whole pizza. In probability, the 'whole pizza' is always represented by the number 1.
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled. Let A be the event that the result is a 6. Find P(A').
🔍 Worked example — Standard
In a bag of colored marbles, the probability of picking a red marble is 0.35. Find the probability of not picking a red marble.
🔍 Worked example — Exam-style
A Venn diagram shows two overlapping circles, A and B, inside a universal set Omega. The probability of an outcome being in A is P(A) = 0.6. The probability of an outcome being in B is P(B) = 0.5. If P(A ∩ B) = 0.2, find the probability that an outcome is in neither A nor B, denoted as P(A' ∩ B').
⚠️ Common errors
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🎯 Exam question
Key ideas — Relative frequency
From an experiment:
🧑🏫 Teacher's explanation
Imagine you are flipping a coin. If you flip it 10 times, you might get 7 heads. Does that mean the coin is broken? Not necessarily. It just means that with a small number of tries, luck plays a huge role. This 'experimental' result is what we call relative frequency.
📖 Definitions
🔍 Worked example — Easy
A bag contains red and blue marbles. A marble is drawn and replaced 50 times. The red marble is drawn 12 times. Calculate the relative frequency of drawing a red marble.
🔍 Worked example — Standard
In a game, the probability of winning a prize is 0.35. If the game is played 200 times, calculate the expected frequency of winning a prize.
🔍 Worked example — Exam-style
A biased spinner is spun 500 times. The relative frequency of landing on 'Green' is 0.4. Calculate the expected frequency of landing on 'Not Green'.
⚠️ Common errors
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🎯 Exam question
Key ideas — Expected frequency
expected frequency = P(A) × number of trials
🧑🏫 Teacher's explanation
Imagine you are playing a game of darts. If you are a professional, you might have a 0.8 probability of hitting the bullseye. If you throw 10 darts, you would "expect" to hit it 8 times. However, you won't necessarily hit it exactly 8 times every single time you play; you might hit it 7 or 9 times. This is because probability describes what happens on average over many trials, not a guaranteed outcome for a single set of trials.
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled 60 times. Calculate the expected frequency of rolling a 4.
🔍 Worked example — Standard
In a large jar of sweets, the probability of picking a red sweet is 0.15. If 200 sweets are picked at random, calculate the expected frequency of picking a red sweet.
🔍 Worked example — Exam-style
A weather forecast states there is a 30\% chance of rain tomorrow. Based on this, how many rainy days would you expect in a period of 365 days? Give your answer to the nearest whole number.
⚠️ Common errors
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🎯 Exam question
Worked examples
A spinner lands on red, blue, green or yellow with probabilities 0.3, 0.25, x and 2x. 0.3 + 0.25 + 3x = 1 ⇒ x = 0.15, so P(yellow) = 0.3.
🔍 Worked example — Easy
A bag contains red, blue, and green marbles. The probability of picking a red marble is 0.4 and the probability of picking a blue marble is 0.35. Find the probability of NOT picking a red marble.
🔍 Worked example — Standard
In an experiment, a spinner is spun 150 times. The number 4 appears 33 times. Calculate the relative frequency of the spinner landing on 4.
🔍 Worked example — Exam-style
A spinner has 5 sections: Red, Blue, Green, Yellow, and White. The probabilities are P(Red) = 0.1, P(Blue) = 0.2, P(Green) = x, P(Yellow) = 2x, and P(White) = 0.3. Find the value of x and use it to calculate the expected frequency of landing on Green in 500 spins.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 2/11. (2) 0.93. (3) 0.58; well above 0.5 over many throws, so it may be biased. (4) 544.