The probability scale and language of likelihood; calculating theoretical probability; estimating probability from data using relative frequency; experimental probability tending to theoretical as trials increase.

What this dot point is asking

Edexcel codes 3p.01, 3p.02 and 3p.06 require you to use the probability scale and the language of likelihood, to calculate theoretical probability, to estimate probability from collected data using relative frequency, and to understand that experimental probability tends towards theoretical probability as the number of trials increases. This is the foundation for every later probability topic, including the distributions.

The probability scale and likelihood

You should be able to place events on the scale and use the language of likelihood: impossible, very unlikely, unlikely, evens, likely, very likely, certain. The probabilities of all the possible outcomes of an event add up to $1$, which gives the complement rule: $P(\text{not } A) = 1 - P(A)$.

Theoretical probability

When all outcomes are equally likely, the theoretical probability of an event is

For example, a fair six-sided dice has $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$. This only works when the outcomes really are equally likely; for a biased dice or a real-world event you cannot count outcomes this way, so you estimate from data instead.

Relative frequency

This is how Edexcel expects you to estimate probabilities for biased or real situations: roll the dice many times, record the results, and divide. Relative frequency is the bridge between collected data (the rest of the course) and probability.

Experimental tends to theoretical

A key idea (code 3p.06) is that as the number of trials increases, the relative frequency gets closer to the true probability, provided the trials are random. A handful of trials can give a misleading estimate because of random variation, but thousands of trials average this out. This is why a larger experiment gives a more reliable estimate, and why you can use a long run of data to estimate the probability of a biased event with confidence.

If you plot the relative frequency against the number of trials, the graph swings about wildly at first and then settles down towards a horizontal level: that level is the estimate of the true probability. Edexcel may show you such a graph and ask what the probability is converging to, or ask you to read off the most reliable estimate (the value at the largest number of trials). The same principle explains why estimates of population parameters improve with larger samples, linking probability back to the rest of the course.

Mutually exclusive outcomes and the total

For a single event, the probabilities of all the separate outcomes must add up to $1$, because one of them is certain to happen. This lets you find a missing probability: if a spinner can land on red, blue or green with $P(\text{red}) = 0.5$ and $P(\text{blue}) = 0.2$, then $P(\text{green}) = 1 - 0.5 - 0.2 = 0.3$. It also underpins the complement rule, and it is the quickest route to "at least one" style questions, where $P(\text{at least one}) = 1 - P(\text{none})$. Checking that your probabilities sum to $1$ is a fast way to catch arithmetic slips.