Probability

Explanation

• A frequency is a count of how many times an outcome occurs.
• In a frequency table, all the frequencies should add to the total number of trials.
• A frequency tree splits a total into groups: the child branches at every split must add back to their parent.
• Complete a missing branch by subtraction, then use the finished tree to compare counts or calculate relative frequencies.
• When groups have different totals, compare proportions rather than raw frequencies.

Explanation

• A random experiment has an uncertain individual result, even though its long-run behaviour can be predicted.
• If outcomes are equally likely, probability is the number of favourable outcomes divided by the total number of outcomes.
• For $n$ repeated trials, expected frequency is $n\times p$.
• This is a prediction, not a guarantee.
• To judge whether a game is fair, compare each player's chances or calculate the expected winnings after any cost; equal-looking sectors or prizes do not by themselves prove fairness.

Explanation

• Probabilities lie on a scale from $0$ to $1$: $0$ is impossible, $1$ is certain, and $0.5$ means equally likely to happen or not happen.
• Theoretical probability comes from a model, while relative frequency comes from results: $\text{relative frequency}=\dfrac{\text{frequency}}{\text{number of trials}}$.
• In many unbiased trials, relative frequency is expected to settle near the theoretical probability, but random variation means the two values need not be exactly equal.

Explanation

• An exhaustive set covers every possible outcome.
• If those outcomes are mutually exclusive, exactly one occurs, so their probabilities add to $1$.
• For mutually exclusive events, $P(A\text{ or }B)=P(A)+P(B)$.
• An event and its complement are exhaustive and mutually exclusive, so $P(\text{not }A)=1-P(A)$.
• If events overlap, subtract the intersection: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$.

Explanation

• An empirical distribution is based on observed relative frequencies; a theoretical distribution is predicted by a probability model.
• With unbiased, independent trials, a larger sample usually gives more stable relative frequencies that are closer to the theoretical probabilities.
• This is a long-run tendency, not a promise that every larger sample will be closer.
• Increasing sample size reduces random variation, but it does not remove bias caused by an unfair device or poor sampling method.

Explanation

• Systematic enumeration means listing every permitted outcome once in a clear order.
• Use a table or grid when two quantities vary, a tree for successive choices, and a Venn diagram for overlapping sets.
• For a two-set Venn diagram, fill the intersection first, then the regions belonging only to each set, then the outside region.
• A list of possible outcomes does not automatically make them equally likely; probability still depends on how the experiment works.

Explanation

• A possibility space lists every permitted outcome once, often as ordered pairs in a grid.
• If two experiments are independent, with $m$ and $n$ equally likely outcomes, the complete space has $mn$ equally likely ordered pairs.
• Probability is then favourable pairs divided by all pairs.
• Order matters when there is a first and second result: $(a,b)$ and $(b,a)$ are different.
• Do not use favourable outcomes over total outcomes unless the combined outcomes are equally likely.

Explanation

• On a probability tree, multiply probabilities along one route and add the products of alternative, mutually exclusive routes.
• Independent events leave later probabilities unchanged.
• Dependent events change later probabilities because an earlier result changes what remains.
• Without replacement, reduce the total and update the relevant outcome count after each draw; with replacement, the original probabilities repeat.
• Always check whether the context justifies equal or repeated branch probabilities.

Explanation

• Higher tier only.
• A conditional probability restricts the sample space to outcomes satisfying the stated condition.
• In $P(A\mid B)$, read 'the probability of $A$ given $B$': the denominator is the total frequency in $B$, and the numerator is the frequency in both $A$ and $B$.
• Two-way tables, Venn diagrams and expected-frequency trees can all supply these restricted counts.
• The overall total is not the denominator unless it is also the conditioning group.