Listing Outcomes with Diagrams
This topic covers systematic ways to list and visualise all possible outcomes of an experiment, which is the essential first step for calculating any probability without a calculator.
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
- Sample Space Diagrams (Grids/Tables) are best for showing all combinations from two independent events, like rolling two dice. The rows represent outcomes of one event, and columns represent the other.
- Venn Diagrams are used to visualise overlaps between different groups or categories. They clearly show which items belong to one category, another, both, or neither.
- Tree Diagrams are ideal for experiments with sequential stages, like picking two marbles from a bag one after the other. Each path from the start to a final branch represents a unique overall outcome.
- For simpler problems, a direct, systematic list of all possibilities is often the fastest way to define the sample space (e.g., listing outcomes of three coin flips as HHH, HHT, HTH, etc.).
Formulae
P(Event) = (Number of desired outcomes) / (Total number of possible outcomes) To calculate the theoretical probability of an event when all outcomes in the sample space are equally likely.
Definitions
- Set
- A collection of distinct items or numbers, often grouped by a common property, such as {multiples of 3 less than 10}.
- Sample Space
- The complete list of all possible individual outcomes of an experiment. For a standard 6-sided die, this is {1, 2, 3, 4, 5, 6}.
- Intersection (Overlap)
- The items that are members of two or more sets at the same time. In a Venn Diagram, this is the region where the circles overlap.
- Union
- The collection of all items that are in at least one of the sets being considered. In a Venn diagram, this corresponds to the total area covered by the circles.
Worked example
In a class of 30 students, 15 study Physics, 20 study Chemistry, and 8 study neither subject. What is the probability that a student chosen at random studies both Physics and Chemistry?
- 1
First, identify the total number of students who study at least one subject.
This is the total in the class minus those who study neither:
30 - 8 = 22 students - 2
This group of 22 students is made up of those who study only Physics, only Chemistry, and both.
Let the number who study both be 'x'.
- 3
If we simply add the total number of Physics and Chemistry students, we get 15 + 20 = 35.
This number is larger than 22 because it counts the students who do both subjects twice.
- 4
The number who do both ('x') is the difference between this sum and the actual total of students studying at least one subject:
x = 35 - 22 = 13 - 5
So, 13 students study both Physics and Chemistry.
- 6
The probability is the number of students who study both, divided by the total number of students in the class:
13 / 30.
Answer: 13/30
Common mistakes
- ×Forgetting to subtract the overlap when finding the total number of items in two sets. Simply adding the totals for each set will double-count any items that are in both.
- ×Misreading constraints in the question, such as the range of numbers (e.g., 'integers from 1 to 20') or a specific property (e.g., 'even numbers only'). This leads to an incorrect total number of outcomes (the denominator).
- ×Confusing 'only A' with 'A'. In a Venn diagram, 'A' refers to the entire circle, including the overlap, while 'only A' refers to the part of the circle that does not overlap with any other.
No-calculator tips
- ✓Always determine the total number of possible outcomes first. This establishes the denominator for your probability and helps frame the rest of the problem.
- ✓Draw a quick, simple sketch of the appropriate diagram (grid, Venn, or tree) on your paper. Visualising the problem makes it much easier to avoid missing outcomes or double-counting.
- ✓When filling in a Venn diagram, start with the most specific piece of information. This is typically the central overlap region (items in 'A' AND 'B' AND 'C'), then work your way outwards.