Probability

The One Thing

Probability measures the likelihood of an event by comparing favourable outcomes with all possible outcomes. For a finite sample space of equally likely outcomes, use $$P(E)=\frac{n(E)}{n(S)}.$$

Definitions and Results

• Random experiment: An action or process whose exact result cannot be predicted in advance, although all possible results are known.
• Outcome: A possible result of a random experiment, such as getting a head when a coin is tossed.
• Sample space: The set of all possible outcomes of a random experiment, usually denoted by $S$.
• Event: A collection of one or more outcomes from the sample space, usually denoted by $E$.
• Favourable outcomes: The outcomes that satisfy the condition of the event being considered.
• Equally likely outcomes: Outcomes that have the same chance of occurring.
• Certain event: An event that must occur; its probability is $1$.
• Impossible event: An event that cannot occur; its probability is $0$.
• Complementary event: The event that an event $E$ does not occur, written as $E^{\prime }$ or not $E$.
• Elementary event: An event containing exactly one outcome.
• Probability of an event: For a finite sample space $S$ with equally likely outcomes,

$$P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}=\frac{n(E)}{n(S)}.$$

• Range of probability: For any event $E$,

$$0\le P(E)\le 1.$$ Equivalently, probability may be written between $0$ and $1$, or between $0\%$ and $100\%$.

• Sample-space probability:

$$P(S)=1,$$ because some outcome in the sample space must occur.

• Empty-event probability:

$$P(\varnothing )=0,$$ because the empty event contains no possible outcomes.

• Complement rule:

$$P(E^{\prime })=1-P(E).$$ Therefore, $$P(E)+P(E^{\prime })=1.$$

• One coin toss:

$$S=\{H,T\}.$$

• One die roll:

$$S=\{1,2,3,4,5,6\}.$$

• Two coin tosses:

$$S=\{HH,HT,TH,TT\},$$ containing four equally likely outcomes.

• Two dice rolled: The total number of ordered outcomes is

$$6\times 6=36.$$

• Experimental and theoretical probability: Experimental probability may differ from theoretical probability in a small number of trials, but generally approaches the theoretical value as the number of trials increases.

Worked Methods

Method 1: Constructing a sample space

• Identify the random experiment.
• List every possible outcome.
• Check that no possible outcome has been omitted.
• Use the resulting set as the sample space $S$.

Examples:

• For one coin toss:

$$S=\{H,T\}.$$

• For one die roll:

$$S=\{1,2,3,4,5,6\}.$$

• For two coins tossed:

$$S=\{HH,HT,TH,TT\}.$$ The outcomes $HT$ and $TH$ are distinct because their order differs.

• For two dice rolled, each die has $6$ possible outcomes, so the total number of ordered outcomes is

$$6\times 6=36.$$

Method 2: Calculating the probability of an event

• Define the random experiment.
• List or determine the complete sample space $S$.
• State the event $E$.
• Count the favourable outcomes $n(E)$.
• Count the total outcomes $n(S)$.
• Confirm that the outcomes are equally likely.
• Apply

$$P(E)=\frac{n(E)}{n(S)}.$$

• Express the result as a fraction, decimal, or percentage between $0$ and $1$, or between $0\%$ and $100\%$.

For example, when two coins are tossed, the sample space is $$S=\{HH,HT,TH,TT\}.$$ Any event is formed by selecting one or more of these outcomes, and its probability is found by dividing the number of selected favourable outcomes by $4$, provided the four outcomes are treated as equally likely.

Method 3: Using complementary probability

• Identify the event $E$.
• Identify its complement $E^{\prime }$, meaning that $E$ does not occur.
• Use

$$P(E^{\prime })=1-P(E).$$

• Alternatively, use

$$P(E)+P(E^{\prime })=1.$$

This method is particularly useful when the complement is easier to count than the event itself.

Method 4: Interpreting special events

• If the event is the entire sample space $S$, it is a certain event:

$$P(S)=1.$$

• If the event is the empty event $\varnothing$, it is an impossible event:

$$P(\varnothing )=0.$$

• If an event contains exactly one outcome, it is an elementary event.

Where It Goes Wrong

• The sample space is not listed completely, so possible outcomes are omitted.
• Outcomes whose order matters are treated as identical; in two coin tosses, $HT$ and $TH$ are different outcomes.
• Favourable outcomes are counted incorrectly because the event has not been described precisely.
• The formula

$$P(E)=\frac{n(E)}{n(S)}$$ is used without checking that all outcomes are equally likely.

• A probability is written outside the required range $0\le P(E)\le 1$, or outside $0\%$ to $100\%$.
• The complement rule is applied incorrectly; the required relationship is

$$P(E^{\prime })=1-P(E),$$ and $P(E)+P(E^{\prime })=1$.

What Gets Asked

• Define a random experiment, outcome, sample space, event, favourable outcome, equally likely outcome, certain event, impossible event, complementary event, or elementary event.
• List the sample space for one coin toss, one die roll, or two coins tossed.
• Determine the number of ordered outcomes when two dice are rolled.
• Identify the favourable outcomes for a stated event.
• Calculate $P(E)$ using

$$P(E)=\frac{n(E)}{n(S)}$$ when outcomes are equally likely.

• Find the probability of a complementary event using

$$P(E^{\prime })=1-P(E).$$

• Explain why $P(S)=1$ and $P(\varnothing )=0$.
• Check whether a proposed probability is valid.
• Distinguish between theoretical probability and experimental probability, including the tendency of experimental probability to approach the theoretical value as the number of trials increases.