Probability and Expected Frequency

The 0 to 1 Probability Scale

Probability is the numerical measure of the likelihood that a particular event will occur. This measure is standardised on a scale from 0 to 1. A probability of 0 indicates that an event is impossible, while a probability of 1 indicates that an event is absolutely certain to happen.

All other probabilities are expressed as values between these two extremes. A probability of $0.5$ (equivalent to $\frac{1}{2}$ or $50\%$) suggests that an event is just as likely to occur as it is not to occur. In the TMUA, probabilities are usually expressed as fractions in their simplest form or as decimals. It is important to remember that a probability can never be negative and can never exceed 1.

Theoretical Probability

Theoretical probability is determined by mathematical reasoning rather than by conducting experiments. It assumes that the physical conditions of an event are known and that all possible outcomes in the sample space are equally likely. For any event $A$, the theoretical probability is defined as:

$P(A) = \frac{\text{Number of outcomes favourable to } A}{\text{Total number of possible outcomes}}$

For example, if a fair six-sided die is rolled, the sample space is $\{1, 2, 3, 4, 5, 6\}$. To find the probability of rolling a number greater than 4, we identify the favourable outcomes, which are $\{5, 6\}$. Thus, $P(x > 4) = \frac{2}{6} = \frac{1}{3}$.

Relative Frequency

Relative frequency, also referred to as experimental probability, is calculated based on the actual results of an experiment or a series of observations. It represents the proportion of times an event has occurred in a given number of trials. The formula is:

$\text{Relative Frequency} = \frac{\text{Observed frequency of the event}}{\text{Total number of trials performed}}$

If a coin is tossed 200 times and lands on heads 104 times, the relative frequency of heads is $\frac{104}{200} = 0.52$. While the theoretical probability of a fair coin is $0.5$, the relative frequency provides an empirical measure that may deviate due to random variation in a finite sample.

Expected Frequency

The expected frequency is the number of times we predict an event will occur if an experiment is repeated a specific number of times. This prediction is based on the event's theoretical probability. If an event has a probability $p$ and we conduct $n$ independent trials, the expected frequency $E$ is given by:

$E = n \times p$

Consider a bag containing 2 red marbles and 8 blue marbles. The theoretical probability of drawing a red marble is $P(\text{red}) = \frac{2}{10} = 0.2$. If we draw a marble and replace it 400 times, the expected frequency of drawing a red marble is:

$E = 400 \times 0.2 = 80$

While the actual number of red marbles drawn in an experiment might be 78 or 83, 80 remains our best theoretical prediction for the outcome.

The Relationship Between Frequency and Probability

The Law of Large Numbers provides the critical link between these concepts. It states that as the number of trials increases, the observed relative frequency will tend to get closer and closer to the theoretical probability.

In the TMUA, you may be presented with experimental data and asked to evaluate a theoretical model. If a very large number of trials shows a relative frequency that is significantly different from the theoretical probability, this may suggest that the initial assumptions (such as the fairness of a die) are incorrect. Conversely, in a small number of trials, large deviations between relative frequency and theoretical probability are common and do not necessarily imply bias.