Calculating Expected Outcomes
Theoretical probability predicts the average outcome of an experiment if it is repeated many times, though this 'expected' outcome is a statistical average, not a guarantee of what will happen in any single set of trials.
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 expected number of times an event will occur is calculated by multiplying its probability by the total number of trials.
- A 'fair' object, like a die or coin, is one where every possible outcome is equally likely.
- The probability of an event is the number of ways that event can happen divided by the total number of possible outcomes.
- The calculated expected outcome is a long-term average. In practice, random variation means the actual results of an experiment can be different.
- For compound events (e.g., rolling an even number), sum the probabilities of the individual successful outcomes (e.g., P(2) + P(4) + P(6)).
Formulae
Expected Occurrences = P(Event) × n To calculate the predicted number of times a specific event will happen over 'n' trials.
Definitions
- Sample Space
- The complete set of all possible outcomes of an experiment. For a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
- Fairness
- A condition in an experiment where each elementary outcome has an equal chance of occurring. For example, an unweighted coin is fair because P(Heads) = P(Tails).
- Expected Outcome
- A theoretical value representing the average result of an experiment if it were repeated an infinite number of times. It is not a prediction for a single experiment.
Worked example
A fair 12-sided die, with faces numbered 1 to 12, is rolled 180 times. What is the expected number of times it will land on a prime number?
- 1
First, identify the total number of possible outcomes for a single roll, which is 12.
- 2
Next, identify the successful outcomes, which are the prime numbers between 1 and 12.
These are 2, 3, 5, 7, and 11.
(Note:
1 is not a prime number).
- 3
There are 5 successful outcomes (prime numbers).
- 4
Calculate the probability of rolling a prime number in a single roll:
P(prime) = (Number of primes) / (Total faces) = 5/12 - 5
Use the formula for expected occurrences:
Expected = P(prime) × number of trials - 6
Substitute the values:
Expected = (5/12) × 180 - 7
Calculate the result:
(180 / 12) × 5 = 15 × 5 = 75.
Answer: 75
Common mistakes
- ×Mistaking the 'expected' value for a guaranteed or exact result. Random chance means the actual number will likely differ.
- ×Incorrectly identifying the number of successful outcomes, for example by miscounting prime numbers or forgetting the conditions of the event.
- ×Making arithmetic errors when multiplying the probability fraction by the number of trials, especially with larger numbers.
No-calculator tips
- ✓Always simplify the probability fraction before multiplying it by the number of trials. For example, use 1/3 instead of 4/12 to make multiplication simpler.
- ✓When multiplying a fraction by a whole number (e.g., (5/12) × 180), it's often easier to do the division first (180 / 12 = 15) and then multiply the result (15 × 5 = 75).