Theoretical Probability and Sample Spaces
This topic covers how to list all possible outcomes of simple and combined experiments (like rolling dice) in a structured way to calculate the probability of a specific event occurring, assuming all outcomes are equally likely.
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
- The fundamental principle is P(Event) = (Number of successful outcomes) / (Total number of possible outcomes).
- For combined experiments, the total number of outcomes is the product of the outcomes for each part. E.g., two dice have 6 × 6 = 36 outcomes.
- A possibility space grid or table is the most reliable method for visualizing all outcomes of a combined experiment, such as the sum or product of two dice rolls.
- Be precise about the set of numbers you are considering. Know your primes (2, 3, 5, 7⋯), squares (1, 4, 9⋯), and factors for a given number.
- Probabilities are always between 0 and 1. An answer greater than 1 or less than 0 is incorrect.
Formulae
P(Event) = (Number of favourable outcomes) / (Total number of outcomes) Use this core formula whenever an experiment has a set of equally likely outcomes.
Definitions
- Possibility Space
- The complete set of all possible, distinct outcomes of an experiment. Also known as a Sample Space.
- Event
- A specific outcome or a collection of outcomes of interest. For example, rolling an even number on a die.
- Equally Likely Outcomes
- A situation where each possible outcome of an experiment has the exact same chance of occurring, like in a fair coin toss or die roll.
Worked example
Two fair six-sided dice are rolled. What is the probability that the product of the two scores is a square number greater than 10?
- 1
First, establish the total number of outcomes.
Each die has 6 faces, so for two dice, the total number of combinations is 6 × 6 = 36.
- 2
Next, identify the event conditions:
the product must be a square number, and it must be greater than 10.
- 3
List the possible products that are square numbers.
The maximum product is 6 × 6 = 36.
The square numbers in this range are 1, 4, 9, 16, 25, 36.
- 4
Apply the second condition:
'greater than 10'.
This leaves us with the target products of 16, 25, and 36.
- 5
Now find the dice combinations that result in these products:
Product 16:
(4, 4).
Product 25:
(5, 5).
Product 36:
(6, 6).
- 6
Count the number of favourable outcomes.
There are 3 combinations:
(4,4), (5,5), and (6,6).
- 7
Calculate the probability:
P = Favourable / Total = 3 / 36 - 8
Simplify the fraction to its lowest terms:
3 / 36 = 1 / 12
Answer: 1/12
Common mistakes
- ×Incorrectly calculating the total number of outcomes for combined events by adding instead of multiplying (e.g., 6+6=12 for two dice instead of 6*6=36).
- ×Failing to read constraints carefully, such as 'greater than' vs 'greater than or equal to', or missing a condition entirely from a multi-part question.
- ×Making errors with number properties, for example, incorrectly identifying 1 as a prime number or missing some of the factors of a number.
No-calculator tips
- ✓Always simplify your final fraction. Working with 1/4 is much easier than 9/36 if a subsequent calculation is needed.
- ✓For problems involving sums or products of two dice, quickly sketch a 6x6 grid. You don't always need to fill it all in, but it helps visualize the 36 outcomes and avoid miscounting.
- ✓If counting the 'successful' outcomes is complicated, try counting the 'unsuccessful' outcomes and subtracting from the total. The probability is 1 - P(unsuccessful).