- Probability is the number of favourable outcomes divided by the total number of equally likely outcomes.
- Every probability is between 0 (impossible) and 1 (certain).
- The probabilities of all possible outcomes add up to 1.
- P(not A) equals 1 minus P(A).
Watch out: Writing a probability as a raw count, like "5 counters", instead of as a fraction of the total number of outcomes.
- 0 means impossible, 1 means certain, and 0.5 means an even chance.
- Write probabilities as fractions, decimals or percentages, never as ratios or as 1 in 5.
- Words like unlikely and likely map to positions on the 0 to 1 scale.
- An event and its opposite sit the same distance from each end of the scale.
Watch out: Describing a small but non-zero probability, such as 0.1, as "impossible" instead of "unlikely", since impossible only applies to a probability of exactly 0.
- Relative frequency is the number of times something happened divided by the number of trials.
- It is an estimate of probability from an experiment, also called experimental probability.
- The more trials you do, the closer relative frequency gets to the true probability.
- Expected frequency is probability multiplied by the number of trials.
Watch out: Dividing the total number of trials by the frequency of the event, instead of the event's frequency by the total number of trials.
- Mutually exclusive events cannot both happen at the same time.
- For mutually exclusive events, P(A or B) equals P(A) plus P(B).
- If a set of outcomes is exhaustive and mutually exclusive, their probabilities sum to 1.
- Rolling a 2 and rolling an odd number on one die are mutually exclusive.
Watch out: Assuming two events are mutually exclusive without checking whether an outcome could satisfy both, for example missing that 6 is both a multiple of 3 and an even number.
- Independent events do not affect each other probability.
- For independent events, P(A and B) equals P(A) times P(B): multiply along the branches.
- Drawing with replacement keeps events independent; drawing without replacement does not.
- Multiply for "and", add for "or".
Watch out: Assuming events are independent without checking whether one outcome affects the other, for example missing that picking without replacement changes the probabilities.
- Each set of branches must have probabilities that add to 1.
- Multiply along the branches to get the probability of a whole path.
- Add the results of the different paths that satisfy the condition.
- Without replacement, the second set of probabilities changes because one item has been removed.
Watch out: Forgetting that the probabilities on branches from the same point must add up to 1 when finding a missing value.
- The overlap is "A and B" (the intersection); everything inside either circle is "A or B" (the union).
- Anything outside both circles is in neither set.
- Fill the intersection first, then the rest of each circle, then the region outside.
- P(A given B) uses only the part of the diagram inside B.
Watch out: Confusing the union symbol, ∪, meaning "or" (everything in either set), with the intersection symbol, ∩, meaning "and" (only the overlap).
- Conditional probability is the chance of one event given that another has already happened.
- P(A given B) equals P(A and B) divided by P(B): restrict yourself to the outcomes where B is true.
- "Without replacement" problems are conditional, because the second probability depends on the first.
- On a tree diagram, the second set of branches shows conditional probabilities.
Watch out: Dividing by the overall total instead of the total for the given condition when finding a conditional probability from a Venn diagram or table.
- A two-way table sorts data by two categories at once, and the totals must agree both ways.
- Fill in the totals first, then find the missing cells by subtraction.
- Read a probability straight from the table: the matching cell over the relevant total.
- Check every row and column adds to its total before you answer.
Watch out: Subtracting from the wrong row or column total when finding a missing value, instead of using the total that contains the missing cell.
