Probability and Expected Outcomes

In this section, we explore the fundamental ideas of probability required for the TMUA. While many students are familiar with basic probability, university admissions tests often require a deeper conceptual understanding of why certain outcomes are expected and how randomness behaves over multiple trials.

Randomness and Fairness

In mathematics, an experiment is considered random if its individual outcomes are uncertain, even though the distribution of outcomes over many repetitions follows a predictable pattern.

Fairness is a specific condition of randomness. When we say a die is 'fair' or a coin is 'unbiased', we are making an assumption of symmetry. In a fair experiment, every possible elementary outcome has an identical probability of occurring. If a fair six-sided die is rolled, the symmetry of the cube ensures that the probability of landing on any one face is exactly $1/6$.

The Principle of Equally Likely Events

The foundation of calculating probabilities in simple experiments is the principle of equally likely events. If we can identify a set of outcomes that are all equally probable, the probability of an event $E$ is defined as:

$P(E) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}$

Worked Example: The Fair Spinner

Consider a fair spinner divided into 8 equal sectors, numbered 1 to 8. We want to find the probability of spinning a prime number.

1. Identify the total number of equally likely outcomes. Here, there are 8 sectors, so $n = 8$.
2. List the successful outcomes (prime numbers). These are 2, 3, 5, and 7. There are 4 such outcomes.
3. Apply the formula: $P(\text{Prime}) = \frac{4}{8} = 0.5$.

Calculating Expected Outcomes

When we repeat a random experiment multiple times, we can calculate the expected outcome. This is a theoretical value representing the average result we would expect if the experiment were repeated an infinite number of times.

For $N$ independent trials of an experiment where the probability of success is $p$, the expected number of successes is:

$E = N \times p$

Worked Example: Multiple Experiments

Suppose we roll a fair six-sided die 120 times. How many times would we expect to roll a 5 or a 6?

1. Calculate the probability of success in a single trial. The successful outcomes are 5 and 6, so $P(5 \text{ or } 6) = \frac{2}{6} = \frac{1}{3}$.
2. Multiply the probability by the number of trials: $E = 120 \times \frac{1}{3} = 40$.

We would expect to see a 5 or a 6 approximately 40 times.

Variability and Repetition

It is vital to understand that the expected outcome is not a guaranteed result. If you were to actually roll a die 120 times, you might roll a 5 or 6 exactly 40 times, but you might also roll them 38 times, 45 times, or 32 times.

This is the nature of randomness: if an experiment is repeated, the outcome may be different. While the theoretical probability remains constant, the actual results of a sample will fluctuate. However, as the number of trials increases, the relative frequency of an outcome (the proportion of times it occurs) generally tends to get closer to the theoretical probability. In the TMUA, you may be asked to identify which statements about these outcomes are 'must be true' versus 'could be true'. Usually, any specific numerical result for a small number of trials 'could' happen, but very few are 'certain' to happen.