Introduction to Probability

The One Thing

Probability measures how likely an event is to occur. For equally likely outcomes, it is calculated by comparing the number of favourable outcomes with the total number of outcomes.

Definitions and Results

• Random experiment: An activity or process whose exact result cannot be predicted in advance, although all possible results can be listed.
• Outcome: A single 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 represented by $S$. For a coin toss, $S=\{H,T\}$. For one roll of a die, $S=\{1,2,3,4,5,6\}$.
• Event: A collection of one or more outcomes from the sample space, usually represented by $E$.
• Favourable outcomes: The outcomes that satisfy the condition of a particular event.
• Equally likely outcomes: Outcomes that have the same chance of occurring, such as heads and tails for a fair coin.
• Probability: A numerical measure of the chance that an event will occur.
• Impossible event: An event that cannot occur; $P(\varnothing )=0$.
• Sure event: An event that must occur; $P(S)=1$.
• Complementary event: The event that an event does not occur, written as $\text{not }E$ or $E^{\prime }$.
• Classical probability formula: When the outcomes are equally likely,

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

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

$$0\le P(E)\le 1.$$

• Complement rule:

$$P(\text{not }E)=1-P(E).$$ Therefore, the probabilities of an event and its complement always add up to $1$.

• A probability may be expressed as a fraction, decimal, or percentage.
• If an event has no favourable outcomes, its probability is $0$; if all outcomes are favourable, its probability is $1$.
• A larger probability means that an event is more likely to occur, but it does not guarantee that the event will occur in a single trial.

Worked Methods

Identifying the sample space

• Identify the random experiment.
• List every possible outcome.
• Write the set of outcomes as the sample space $S$.

For a coin toss: $$S=\{H,T\}.$$

For one roll of a die: $$S=\{1,2,3,4,5,6\}.$$

Finding the probability of an event

• State the sample space.
• Identify the event $E$.
• Count the favourable outcomes: those satisfying the event’s condition.
• Count the total outcomes in the sample space.
• Check that the outcomes are equally likely.
• Apply

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

• Express the answer as a fraction, decimal, or percentage if required.

The classical formula is used only when the outcomes are equally likely. For example, heads and tails are equally likely for a fair coin.

Using the complement of an event

• Identify the event $E$.
• Identify its complementary event, written as $\text{not }E$ or $E^{\prime }$.
• Use

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

• Interpret the result as the probability that $E$ does not occur.

Checking a probability

• Verify that the result satisfies

$$0\le P(E)\le 1.$$

• Check that the sample space contains all possible outcomes.
• Check that the favourable outcomes are selected from the sample space.
• Confirm that the classical formula was used only for equally likely outcomes.
• If no outcomes are favourable, the probability should be $0$; if every outcome is favourable, it should be $1$.

Where It Goes Wrong

• The sample space is incomplete: all possible outcomes must be listed before favourable outcomes are counted.
• Favourable outcomes are confused with total outcomes; the numerator counts only outcomes satisfying the event, while the denominator counts all outcomes in the sample space.
• The classical formula is applied when outcomes are not equally likely; it requires equally likely outcomes.
• The complement is calculated incorrectly; the required relationship is $P(\text{not }E)=1-P(E)$.
• Probability is treated as certainty: except for impossible and sure events, it describes likelihood rather than guaranteeing the result of a single trial.
• The answer falls outside the required range $0\le P(E)\le 1$, indicating an error in identifying or counting outcomes.

What Gets Asked

• Define a random experiment, outcome, sample space, event, favourable outcomes, equally likely outcomes, probability, impossible event, sure event, or complementary event.
• List the sample space for a coin toss or one roll of a die.
• Identify the favourable outcomes for a stated event.
• Calculate $P(E)$ using the classical formula for equally likely outcomes.
• Determine the probability of an impossible or sure event.
• Find the probability of a complementary event using $P(\text{not }E)=1-P(E)$.
• Express a probability as a fraction, decimal, or percentage.
• Check whether a proposed probability is valid using $0\le P(E)\le 1$.
• Explain why a larger probability indicates greater likelihood but does not guarantee occurrence in one trial.