Probability

The One Thing

Probability measures the likelihood of an event on a scale from 0 to 1, or from 0% to 100%. Addition combines alternative outcomes, whereas multiplication combines events occurring together or in sequence; the correct method depends on whether events are mutually exclusive, independent or dependent.

Definitions and Results

• Probability: A number describing the likelihood of an event. It satisfies $0\le P(A)\le 1$, and may be written as a fraction, decimal or percentage.
• Probability scale: A scale from 0 to 1, where 0 means impossible, 1 means certain, and intermediate values indicate different levels of likelihood.
• Outcome: A possible result of an experiment, such as rolling a 4 on a die.
• Event: A collection of one or more outcomes, such as rolling an even number.
• Sample space: The complete set of all possible outcomes of an experiment. The probabilities of all outcomes in a complete sample space add to 1.
• Complement: The event that an event does not occur. It is written $A^{\prime }$ or “not $A$”, with

$$P(\text{not }A)=1-P(A).$$

• Equally likely outcomes: If all outcomes are equally likely,

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

• Mutually exclusive events: Events that cannot happen at the same time, such as rolling a 2 and rolling a 5 on one roll of a die. For mutually exclusive events,

$$P(A\text{ or }B)=P(A)+P(B).$$

• Independent events: Events for which the result of one event does not affect the probability of the other. For independent events,

$$P(A\text{ and }B)=P(A)\times P(B).$$

• Conditional probability: The probability of one event when another event is already known to have happened. For dependent events, later probabilities may change after an earlier outcome, especially when objects are not replaced.
• Addition rule for any two events:

$$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B).$$

• Expected frequency: The number of times an event is predicted to occur in a specified number of trials:

$$\text{Expected frequency}=\text{probability of the event}\times \text{number of trials}.$$ It is a prediction, so it need not be a whole number or equal to the actual frequency.

• Sample-space diagram: A diagram listing all possible combinations of outcomes, often for two events. When two fair dice are rolled, there are 36 equally likely ordered outcomes.
• Tree diagram: A branching diagram showing sequences of events, with probabilities written on the branches. Probabilities are multiplied along a path and suitable path probabilities are added.
• Venn diagram: A diagram showing relationships between sets or events.
• Intersection: The outcomes common to two events, written $A\cap B$, meaning “$A$ and $B$”.
• Union: All outcomes in either event or in both, written $A\cup B$, meaning “$A$ or $B$”. In a Venn diagram, the intersection represents “and”, while the combined regions represent “or”.
• Probability model: A valid model has every probability between 0 and 1, and the probabilities of all possible outcomes add to 1.
• Fair die and fair coin: For a fair die, each face has probability $\frac{1}{6}$; for a fair coin, each side has probability $\frac{1}{2}$.

Worked Methods

1. Finding a probability from equally likely outcomes

• Identify the complete sample space.
• Count the outcomes favourable to the event.
• Count the total number of possible outcomes.
• Use

$$P(A)=\frac{\text{favourable outcomes}}{\text{total outcomes}}.$$

For a fair die, each face has probability $\frac{1}{6}$. The outcome “rolling a 4” therefore has probability $\frac{1}{6}$. An event such as “rolling an even number” consists of the outcomes $2,4,6$.

2. Using complements

• Identify the event $A$.
• Consider the complementary event, “not $A$”.
• Subtract the probability of $A$ from 1:

$$P(\text{not }A)=1-P(A).$$

The complement is used when calculating the event that $A$ does not occur.

3. Combining mutually exclusive events

• Check that the events cannot occur at the same time.
• Add their probabilities:

$$P(A\text{ or }B)=P(A)+P(B).$$

For example, rolling a 2 and rolling a 5 on one roll of a die are mutually exclusive events, since one roll cannot produce both outcomes.

4. Combining non-mutually exclusive events

• Add $P(A)$ and $P(B)$.
• Subtract the probability of the intersection, because it has been counted twice:

$$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B).$$

• Interpret $A\cap B$ as the outcomes common to both events and $A\cup B$ as all outcomes in either event or both.

5. Combining independent events

• Check that the result of the first event does not affect the second.
• Multiply the probabilities:

$$P(A\text{ and }B)=P(A)\times P(B).$$

For a fair coin, each side has probability $\frac{1}{2}$. For independent coin events, the probability of a specified outcome on two trials is found by multiplying the relevant $\frac{1}{2}$ probabilities.

6. Using a sample-space diagram

• List every possible outcome.
• For two fair dice, construct the 36 equally likely ordered outcomes.
• Identify the outcomes satisfying the required event.
• Divide the number of favourable outcomes by 36.

The ordered outcomes must all be included, because a sample-space diagram represents the complete set of possible results.

7. Using a tree diagram

• Draw a branch for each possible first outcome.
• Write the probability of each first outcome on its branch.
• From each first branch, draw branches for the possible second outcomes.
• If the events are independent, retain the same probabilities on later branches.
• If the events are dependent, update the later branch probabilities after the earlier outcome, particularly when objects are not replaced.
• Multiply probabilities along each path to find the probability of that sequence.
• Add the probabilities of all suitable paths to find an event represented by several paths.

8. Calculating expected frequency

• Identify the probability of the event.
• Identify the number of trials.
• Multiply:

$$\text{Expected frequency}=\text{probability}\times \text{number of trials}.$$

• Interpret the result as a prediction rather than a guaranteed actual frequency.

The expected frequency may not be a whole number, and experimental results may differ because of random variation.

9. Checking a probability model

• Check that no probability is negative.
• Check that no probability is greater than 1.
• Check that the probabilities of all possible mutually exclusive and exhaustive outcomes add to 1.

A model failing any of these checks is not a valid probability model.

Where It Goes Wrong

• Adding probabilities for events that occur together or in sequence: multiplication is required for independent combined events.
• Multiplying probabilities for alternative outcomes: addition is required for mutually exclusive alternatives.
• Using $P(A)+P(B)$ for overlapping events without subtracting $P(A\text{ and }B)$.
• Treating dependent events as independent: later branch probabilities may change, especially when objects are not replaced.
• Omitting ordered outcomes in the 36-outcome sample-space diagram for two fair dice.
• Treating expected frequency as an exact actual result; it is a prediction and may be non-integral.

What Gets Asked

• Place probabilities on a probability scale and express them as fractions, decimals or percentages.
• Find the probability of an event from equally likely outcomes.
• Use a complement to calculate the probability that an event does not occur.
• Combine mutually exclusive events using addition.
• Apply the general addition rule to overlapping events.
• Calculate the probability of independent events using multiplication.
• Use conditional probabilities when events are dependent or objects are not replaced.
• Complete or interpret a sample-space diagram, including the 36 ordered outcomes for two fair dice.
• Use a tree diagram to calculate probabilities along paths and across multiple suitable paths.
• Interpret intersections and unions in a Venn diagram.
• Calculate expected frequency from a probability and a number of trials.
• Determine whether a probability model is valid by checking individual probabilities and their total.