How to Calculate Probability for Single Events in IGCSE Maths
Understanding how to calculate the probability of single events is a core requirement in IGCSE Maths. This page explains how probability is measured on a scale from impossible to certain, and focuses on theoretical probability, experimental probability, and expected frequency using clear exam style examples.
What is Probability?
Probability measures how likely an event is to happen. Every probability is a number between 0 and 1 (inclusive). An event that is impossible has a probability of 0, and an event that is certain has a probability of 1.
Core Ideas
\(0 \leq P(A) \leq 1\) for any event \(A\). A probability of 0 means impossible; 1 means certain. Probabilities can be written as fractions, decimals, or percentages.
When all outcomes are equally likely, calculate probability by counting favourable outcomes and dividing by the total.
When outcomes are not equally likely (or you want to test), use relative frequency from actual trials. More trials give a better estimate.
The probability of an event not happening is \(P(A') = 1 - P(A)\). The event either happens or it doesn't; these two probabilities always add to 1.
Theoretical Probability
When all outcomes are equally likely, the probability of event \(A\) is:
💡 Example 1: Spinner
A fair spinner has five equal sections coloured Red, Blue, Green, Yellow and White. Find the probability of landing on Blue.
There are 5 equally likely outcomes. Blue is 1 of them.
\[P(\text{Blue}) = \frac{1}{5}\]💡 Example 2: Picking a number
A bag contains 12 tickets numbered 1 to 12. One ticket is drawn at random. Find the probability that the number drawn is a multiple of 4.
The multiples of 4 from 1 to 12 are 4, 8, and 12 (three favourable outcomes out of twelve).
\[P(\text{multiple of }4) = \frac{3}{12} = \frac{1}{4}\]The Complement: P(not A)
If event \(A\) either happens or does not happen, and nothing else is possible, then:
\(A'\) means "not \(A\)". This is useful when it is easier to count what you don't want.
💡 Example 3: Complement of a die roll
A fair die is rolled. Find the probability of not rolling a 4.
💡 Example 4: Cards
One card is taken at random from a well-shuffled deck of 52 playing cards. Work out the probability that the card is not a heart.
There are 13 hearts in the deck.
\[P(\text{heart}) = \frac{13}{52} = \frac{1}{4}\] \[P(\text{not a heart}) = 1 - \frac{13}{52} = \frac{39}{52} = \frac{3}{4}\]Experimental Probability (Relative Frequency)
When you carry out an experiment or collect data from real observations, the experimental probability (also called relative frequency) is:
The more trials you carry out, the closer the experimental probability gets to the theoretical probability. This is why experiments with 10 trials give rough estimates, but experiments with 500 trials give much better ones.
💡 Example 5: Biased spinner
A spinner is spun 200 times. It lands on red 72 times, blue 53 times, and green 75 times. Estimate the probability of landing on blue.
This is an estimate based on the experiment. The spinner is likely biased because the frequencies are not equal.
Expected Frequency
If you know the probability of an event and the number of trials, you can predict how many times the event should occur:
The expected frequency is a prediction, not a guarantee. The actual result may differ, but over a large number of trials the actual frequency should be close to the expected frequency.
💡 Example 6: Coin flips
A fair coin is flipped 80 times. How many times would you expect it to land on heads?
You would expect about 40 heads, but the actual number could be slightly higher or lower.
💡 Example 7: Quality control
A factory produces 2,000 light bulbs per day. Testing shows that 1.5% are faulty. How many faulty bulbs are expected each day?
\(P(\text{faulty}) = 0.015\)
\[\text{Expected} = 2000 \times 0.015 = 30\]The factory should expect about 30 faulty bulbs per day.
🔑 Key Points
- Probability is always between 0 and 1 inclusive: \(0 \leq P(A) \leq 1\).
- Theoretical probability uses equally likely outcomes: favourable ÷ total.
- Experimental probability uses results from trials: successes ÷ total trials.
- The complement rule: \(P(A') = 1 - P(A)\). Use it when "not" is easier to count.
- Expected frequency = number of trials \(\times\) probability. It is a prediction, not exact.
- More trials give a more reliable experimental probability.
- Answers can be given as fractions, decimals, or percentages unless the question specifies.
⚠️ Common Mistakes
- Probability greater than 1: if your answer is above 1 or below 0, something has gone wrong.
- Not simplifying fractions: always simplify unless told otherwise (e.g. \(\frac{13}{52} = \frac{1}{4}\)).
- Confusing theoretical and experimental: theoretical uses equally likely outcomes; experimental uses actual trial data.
- Forgetting the complement: \(P(\text{not } A) = 1 - P(A)\), not \(1 \div P(A)\).
- Writing expected frequency as a probability: expected frequency is a whole number (or close to one), not a fraction between 0 and 1.
- Assuming experimental = theoretical: they are only approximately equal, and only with many trials.