Calculating Expected Outcomes
This topic covers how to use theoretical probability to predict the average outcome of an experiment if it's repeated many times. It also emphasizes that 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).