Probability

The One Thing

Probability assigns numbers from 0 to 1 to events in a sample space to represent their likelihood. Its rules follow from three axioms: non-negativity, normalization of the sure event, and additivity for mutually exclusive events.

Definitions and Results

• Random Experiment: A process that can be repeated under similar conditions but whose exact outcome cannot be predicted with certainty.
• Outcome: A possible result of a random experiment, such as obtaining 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 subset of the sample space; it may contain one or more outcomes.
• Simple Event: An event containing exactly one outcome.
• Compound Event: An event containing more than one outcome.
• Sure Event: An event that always occurs; it is the sample space $S$, with $P(S)=1$.
• Impossible Event: An event that cannot occur; it is represented by the empty set $\varnothing$, with $P(\varnothing )=0$.
• Complementary Event: For an event $A$, the complement $A^{\prime }$ consists of all outcomes in $S$ that are not in $A$. It satisfies

$$A\cup A^{\prime }=S,A\cap A^{\prime }=\varnothing ,$$ and $$P(A^{\prime })=1-P(A).$$

• Mutually Exclusive Events: Events that cannot occur together. For events $A$ and $B$,

$$A\cap B=\varnothing .$$ Therefore, $$P(A\cup B)=P(A)+P(B).$$

• Exhaustive Events: A collection of events whose union is the entire sample space.
• Equally Likely Outcomes: Outcomes that have the same chance of occurring. For a finite sample space with equally likely outcomes,

$$P(A)=\frac{n(A)}{n(S)},$$ where $n(A)$ is the number of outcomes in $A$ and $n(S)$ is the total number of outcomes.

• Axiomatic Probability: A system that defines probability through basic rules proposed by A. N. Kolmogorov. The three probability axioms are:

$$P(A)\ge 0\text{for every event }A,$$ $$P(S)=1,$$ and probability is additive for mutually exclusive events.

• Union of Events: $A\cup B$ represents the event that $A$, $B$, or both occur.
• Intersection of Events: $A\cap B$ represents the event that both $A$ and $B$ occur.
• Difference of Events: $A-B$ represents the event that $A$ occurs but $B$ does not. Its probability is

$$P(A-B)=P(A)-P(A\cap B).$$

• Basic Probability Range: For every event $A$,

$$0\le P(A)\le 1.$$ A probability near $1$ indicates that an event is highly likely, whereas a probability near $0$ indicates that it is highly unlikely.

• General Addition Rule: For any two events $A$ and $B$,

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$ The subtraction is required because outcomes common to both events are counted twice in $P(A)+P(B)$.

• Subset Relation: If $A$ is a subset of $B$, then

$$P(A)\le P(B).$$

• Finite Additivity: For mutually exclusive events $A_1,A_2,\dots ,A_n$,

$$P(A_1\cup A_2\cup \cdots \cup A_n)=P(A_1)+P(A_2)+\cdots +P(A_n).$$

Worked Methods

1. Constructing a probability model

• Identify the random experiment.
• List every possible outcome.
• Form the sample space $S$.
• Define the event as a subset of $S$.
• Assign probabilities satisfying the axioms:

- every probability is non-negative; - $P(S)=1$; - probabilities of mutually exclusive events add.

For example, when a coin is tossed, obtaining a head is an outcome. The sample space must contain all possible outcomes of the experiment, and the probabilities assigned to those outcomes must total $1$.

2. Finding a probability from equally likely outcomes

For a finite sample space with equally likely outcomes:

• Count the outcomes in the event, $n(A)$.
• Count all outcomes in the sample space, $n(S)$.
• Apply

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

The event must be defined within the complete sample space, and the outcomes must be equally likely for this formula to apply.

3. Applying the complement rule

• Identify the event $A$.
• Identify its complement $A^{\prime }$, consisting of outcomes in $S$ that are not in $A$.
• Use

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

This method is useful when it is easier to calculate the probability that an event does not occur. Since $$A\cup A^{\prime }=S,A\cap A^{\prime }=\varnothing ,$$ their probabilities sum to $P(S)=1$.

4. Adding mutually exclusive events

• Confirm that the events cannot occur together.
• Check that

$$A\cap B=\varnothing .$$

• Add their probabilities:

$$P(A\cup B)=P(A)+P(B).$$

For several mutually exclusive events, $$P(A_1\cup A_2\cup \cdots \cup A_n)=P(A_1)+P(A_2)+\cdots +P(A_n).$$

5. Applying the general addition rule

• Determine $P(A)$.
• Determine $P(B)$.
• Determine the intersection $P(A\cap B)$.
• Use

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$

The term $P(A\cap B)$ must be subtracted because outcomes in the intersection are counted once in $P(A)$ and once in $P(B)$.

6. Finding the probability of a difference

• Identify the event $A-B$, meaning that $A$ occurs but $B$ does not.
• Determine $P(A)$.
• Determine the probability of the overlapping part, $P(A\cap B)$.
• Apply

$$P(A-B)=P(A)-P(A\cap B).$$

Where It Goes Wrong

• Adding $P(A)+P(B)$ for overlapping events without subtracting $P(A\cap B)$; the general addition rule is required unless the events are mutually exclusive.
• Applying the mutually exclusive addition rule without checking the condition $A\cap B=\varnothing$.
• Forgetting that the sample space must include all possible outcomes and that $P(S)=1$.
• Using

$$P(A)=\frac{n(A)}{n(S)}$$ without confirming that the sample space is finite and its outcomes are equally likely.

• Confusing the complement $A^{\prime }$ with the event $A$; the complement contains outcomes in $S$ that are not in $A$, and $P(A^{\prime })=1-P(A)$.
• Omitting the intersection when calculating a difference: $P(A-B)=P(A)-P(A\cap B)$.

What Gets Asked

This material supports questions requiring students to:

• Define a random experiment, outcome, sample space, event, simple event, compound event, sure event, impossible event, complementary event, mutually exclusive events, exhaustive events, and equally likely outcomes.
• State A. N. Kolmogorov’s three probability axioms.
• Construct a sample space and identify events as subsets of it.
• Calculate probabilities using

$$P(A)=\frac{n(A)}{n(S)}$$ for finite sample spaces with equally likely outcomes.

• Find the probability of a complement using

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

• Apply the addition rule for mutually exclusive events.
• Apply the general addition rule to overlapping events.
• Determine probabilities involving unions, intersections, and differences of events.
• Use the relation $A\subseteq B$ to establish that $P(A)\le P(B)$.
• Explain why $P(A\cap B)$ is subtracted in the general addition rule.
• Interpret probability values near $0$ and $1$ as indicating events that are highly unlikely or highly likely.