Theoretical and Experimental Probability
This topic covers the fundamental principles of probability, focusing on how to calculate the theoretical chance of an event using a simple formula and placing it on the standard 0 to 1 scale. It also connects this theoretical chance to the expected outcomes of real-world experiments.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Probability is a numerical measure of likelihood, ranging from 0 (impossible event) to 1 (certain event).
- Theoretical probability is calculated when all possible outcomes are equally likely, such as in a fair coin toss or die roll.
- The sum of the probabilities of all possible outcomes for any event is always equal to 1.
- Relative expected frequency is a prediction of how many times an event will occur over a set number of trials. It is found by multiplying the event's probability by the number of trials.
- Answers can be expressed as fractions, decimals, or percentages. Always simplify fractions unless told otherwise.
- To find the probability of an event *not* occurring, subtract the probability of it occurring from 1.
Formulae
P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes) To calculate the probability of a single event when all outcomes are equally likely.
Expected Frequency = P(Event) × Number of trials To predict the number of times an event will occur over multiple trials.
Definitions
- Theoretical Probability
- The likelihood of an event calculated based on reasoning and the number of favourable outcomes versus the total possible outcomes, assuming ideal conditions.
- Sample Space
- The complete set of all possible outcomes of an experiment. For a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
- Relative Expected Frequency
- The number of times you would predict an event to happen in a series of trials. It links theoretical probability to experimental predictions.
Worked example
A spinner is divided into 8 equal sectors, numbered 1 to 8. What is the probability that the spinner lands on a prime number? If the spinner is spun 40 times, what is the expected number of times it will land on a square number?
- 1
Identify the sample space, which is {1, 2, 3, 4, 5, 6, 7, 8}.
The total number of outcomes is 8.
- 2
For the first part, identify the prime numbers in the sample space:
{2, 3, 5, 7}.
There are 4 favourable outcomes.
- 3
Calculate the probability of landing on a prime number:
P(Prime) = 4 / 8 = 1/2 - 4
For the second part, identify the square numbers in the sample space:
{1, 4}.
There are 2 favourable outcomes.
- 5
Calculate the probability of landing on a square number:
P(Square) = 2 / 8 = 1/4 - 6
Calculate the expected frequency for 40 spins:
Expected Frequency = P(Square) × 40 = (1/4) × 40.
- 7
The result is 40 / 4 = 10.
Answer: The probability of landing on a prime number is 1/2. The expected number of times it will land on a square number is 10.
Common mistakes
- ×Incorrectly identifying special numbers. Remember that 1 is a square number but is not a prime number, and 2 is the only even prime number.
- ×Making calculation errors when finding the number of outcomes that are 'not' something. It's often safer to find the probability of the event and subtract from 1.
- ×Forgetting to simplify the probability fraction, which makes subsequent calculations (like finding expected frequency) more difficult without a calculator.
No-calculator tips
- ✓When calculating expected frequency, such as (3/5) × 50, perform the division first if possible (50/5 = 10) and then multiply (10 × 3 = 30). This avoids large intermediate numbers.
- ✓To find the probability of something *not* happening, calculate P(event) first and then find 1 - P(event). For example, P(not 6) on a die is 1 - P(6) = 1 - 1/6 = 5/6.
- ✓Quickly convert between common fractions and decimals (e.g., 1/8 = 0.125, 1/5 = 0.2, 1/4 = 0.25) to handle questions that mix formats.