TheIBAxolotl
The vocabulary
- A trial is one run of an experiment (one roll of a die).
- An outcome is a single possible result (rolling a 4).
- The sample space \(U\) is the set of all possible outcomes.
- An event \(A\) is a set of outcomes we are interested in (rolling an even number).
Theoretical probability
When all outcomes are equally likely,
\[P(A) = \frac{n(A)}{n(U)}\]
where \(n(A)\) counts the outcomes in \(A\). Every probability lies between 0 (impossible) and 1 (certain).
Experimental probability
If outcomes are not equally likely, or the situation is too complicated to analyse, run the experiment many times and use the relative frequency:
\[P(A) \approx \frac{\text{number of times } A \text{ occurred}}{\text{number of trials}}\]
The more trials, the closer this tends to be to the true probability. A drawing pin landing point-up is a classic case where only experiment can tell you the probability.
Complementary events
The complement \(A'\) means "\(A\) does not happen". Since one of them must occur,
\[P(A') = 1 - P(A)\]
This is the fastest route for "at least one" questions: \(P(\text{at least one}) = 1 - P(\text{none})\).
Expected number of occurrences
If an event has probability \(p\) and there are \(n\) trials, the expected number of times it happens is \(np\). This is a long-run average and need not be a whole number.
Worked example
Two fair six-sided dice are rolled. The sample space has \(6 \times 6 = 36\) equally likely outcomes, best shown in a grid.
- Sum of 8: \((2,6), (3,5), (4,4), (5,3), (6,2)\), so \(P = \frac{5}{36}\).
- Sum not 8: \(1 - \frac{5}{36} = \frac{31}{36}\).
- In 180 rolls of the pair, the expected number of 8s is \(180 \times \frac{5}{36} = 25\).
Bag example. A bag has 5 red, 7 blue and 8 green counters. One is drawn at random: \(P(\text{blue}) = \frac{7}{20}\) and \(P(\text{not green}) = 1 - \frac{8}{20} = \frac{3}{5}\).
Common mistakes
- Counting \((3,5)\) and \((5,3)\) as the same outcome; with two dice they are different.
- Giving a probability greater than 1 or as a ratio like "5:36".
- Rounding an expected value to a whole number when the question did not ask for it.
- Confusing experimental and theoretical probability in explanations.
Exam tips
- Draw a sample-space grid for two dice or two spinners; it makes counting reliable and earns marks.
- Leave probabilities as exact fractions where possible, or give decimals to 3 significant figures.
- Look for "at least one" and switch to the complement immediately.
- When asked why an experimental result differs from the theoretical one, mention the number of trials and random variation.