Probability

The One Thing

Probability measures the likelihood of events and provides methods for analysing compound chance situations. Correct solutions depend on defining the sample space and selecting the appropriate method—addition, multiplication, diagrams, tables or simulation—according to the relationships between events.

Definitions and Results

• Experiment: A chance process that produces an outcome, such as rolling a die or selecting a card.
• Outcome: A possible result of a probability experiment.
• Sample space: The complete set of all possible outcomes of an experiment.
• Event: One outcome or a collection of outcomes from the sample space.
• Probability: A number between 0 and 1 that describes the likelihood of an event. A probability of 0 means impossible, and a probability of 1 means certain.
• Theoretical probability: The probability calculated from a mathematical model, usually by comparing favourable outcomes with total equally likely outcomes. For equally likely outcomes,

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

• Experimental probability: The probability estimated from repeated trials:

$$\text{Experimental probability}=\frac{\text{frequency of the event}}{\text{total number of trials}}.$$

• Complement: The event that an event does not occur. The complement of $A$ is written $A^{\prime }$ or not $A$:

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

• Mutually exclusive events: Events that cannot occur at the same time. For mutually exclusive events,

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

• Independent events: Events where the occurrence of one does not change the probability of the other:

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

• Dependent events: Events where the occurrence of one changes the probability of the other:

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

• Compound event: An event made up of two or more simpler events.
• Simulation: A model using random numbers, technology or repeated trials to imitate a chance process.
• Expected frequency: The number of times an event is predicted to occur:

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

• General addition rule: For any two events,

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

• Probability bounds: Every probability satisfies

$$0\le P(A)\le 1.$$ The probabilities of all outcomes in a sample space add to 1.

• Tree-diagram rule: A tree diagram represents the stages of a compound experiment. Probabilities are multiplied along a branch and added for separate successful branches.
• Two-way table: A table that organises frequencies or probabilities for two categories. It can be used to find joint, marginal and conditional probabilities.
• Theoretical and experimental probability: As the number of trials increases, experimental probability generally tends to become closer to theoretical probability, although random variation can still occur.
• Simulation results: Simulation results are estimates and may differ from theoretical probability because of sampling variation.
• Replacement: With replacement, an item is returned after selection, usually keeping trials independent. Without replacement, the probabilities often change, making trials dependent.

Worked Methods

Calculating theoretical probability

• Define the experiment and its sample space.
• Identify the event and count its favourable outcomes.
• Confirm that the outcomes are equally likely if using the basic formula.
• Divide the number of favourable outcomes by the total number of outcomes:

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

• Check that the answer lies between 0 and 1.

For example, when the experiment is rolling a die, the outcomes are the possible die results. The sample space must include every possible result before the favourable outcomes are counted.

Using complements

• Identify event $A$.
• Identify the complementary event, not $A$, which means that $A$ does not occur.
• Use

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

• Interpret the result in context.

This method is useful when calculating the probability of an event directly is more difficult than calculating the probability that it does not occur.

Adding probabilities

For mutually exclusive events:

• Confirm that $A$ and $B$ cannot occur at the same time.
• Add their probabilities:

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

For events that may overlap:

• Calculate $P(A)$, $P(B)$, and $P(A\text{ and }B)$.
• Add $P(A)$ and $P(B)$.
• Subtract the overlap:

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

Multiplying probabilities

For independent events:

• Establish that the occurrence of one event does not change the probability of the other.
• Multiply the probabilities:

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

For dependent events:

• Identify the first event.
• Determine how the first event changes the probability of the second.
• Use the conditional probability:

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

• Take account of whether an item is replaced. With replacement, trials usually remain independent; without replacement, trials often become dependent.

Constructing and using a tree diagram

• Identify the stages of the compound experiment.
• Draw branches for the possible outcomes at the first stage.
• Add branches for the possible outcomes at the next stage.
• Label each branch with its probability.
• Multiply probabilities along each branch.
• Add the probabilities of all branches representing the required event.

A tree diagram therefore shows the stages of a compound experiment and distinguishes between separate successful pathways.

Using a two-way table

• Identify the two categories being compared.
• Enter the given frequencies or probabilities in the appropriate cells.
• Calculate row totals, column totals and the overall total where necessary.
• Use the cells to find joint probabilities.
• Use row or column totals to find marginal probabilities.
• Use relevant totals to calculate conditional probabilities.

Calculating experimental probability

• Conduct or examine repeated trials.
• Count the frequency of the event.
• Count the total number of trials.
• Divide:

$$\text{Experimental probability}=\frac{\text{frequency of the event}}{\text{total number of trials}}.$$

• Compare the result with the theoretical probability, allowing for random variation.

As the number of trials increases, the experimental probability generally becomes closer to the theoretical probability, although it may not equal it exactly.

Calculating expected frequency

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

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

• Interpret the result as a prediction rather than a guarantee.

Designing and interpreting a simulation

• Identify the real-world chance situation.
• Define all possible outcomes.
• Assign appropriate probabilities to those outcomes.
• Use random numbers, technology or repeated trials to imitate the chance process.
• Repeat the random process many times.
• Calculate the proportion of simulated trials producing the event.
• Interpret the result in context.
• Compare the estimate with the theoretical probability where one is available.

A simulation is useful when a probability is difficult to calculate directly, but it must accurately represent the original situation. Its results are estimates and can differ from theoretical probability because of sampling variation.

Where It Goes Wrong

• Failing to define the complete sample space before calculating theoretical probability.
• Using the favourable-outcomes formula when outcomes are not equally likely or when the model’s assumptions have not been established.
• Adding probabilities across stages of a compound event instead of multiplying probabilities along a branch.
• Adding probabilities for events that overlap without subtracting $P(A\text{ and }B)$, or adding probabilities for events that are not mutually exclusive.
• Treating dependent events as independent and forgetting that not replacing an item often changes later probabilities.
• Treating experimental or simulated probability as an exact guarantee rather than an estimate affected by random variation and sample size.

What Gets Asked

• Define an experiment, outcome, sample space, event, complement, mutually exclusive events, independent events, dependent events, compound event, simulation or expected frequency.
• List a sample space and calculate a theoretical probability for equally likely outcomes.
• Use the complement rule to find the probability that an event does not occur.
• Calculate probabilities for “or” events using either the mutually exclusive rule or the general addition rule.
• Calculate probabilities for “and” events using independent or dependent multiplication rules.
• Construct or interpret a tree diagram for a compound experiment, including multiplication along branches and addition across successful branches.
• Complete or use a two-way table to find joint, marginal and conditional probabilities.
• Calculate experimental probability from repeated-trial data.
• Calculate expected frequency using probability multiplied by the number of trials.
• Design or interpret a simulation using random numbers, technology or repeated trials.
• Compare theoretical, experimental and simulated probabilities, explaining differences through random variation and sampling variation.
• Interpret a probability in context, recognising that a likely event can fail to occur and an unlikely event can still occur.
• Explain how replacement affects independence and dependence.