Relationships Between Independent Events

In probability, two events are called independent when the occurrence of one has no effect on the likelihood of the other. Recognizing this relationship, and knowing when it does not hold, is essential for solving multi-event probability problems correctly. This lesson formalizes the definition of independence, introduces the multiplication rule for computing joint probabilities, and contrasts independent events with dependent and conditional relationships that frequently appear in exam questions.

Two events \(A\) and \(B\) are independent if knowing that \(B\) has occurred gives no new information about whether \(A\) will occur, and vice versa. Formally, this is written as \( P(A \mid B) = P(A) \) and \( P(B \mid A) = P(B) \). Flipping a coin twice is a classic example: the result of the first flip has no bearing on the second, because the coin has no memory of its previous outcome.

Contrast this with dependent events, where the outcome of one event genuinely changes the probability of the other. Drawing a card from a deck and not replacing it before a second draw is dependent, because removing the first card changes the composition of the remaining deck. Before applying any probability rule, a student should ask whether the mechanism generating the second event has been altered by the first.

Once independence is established, the probability that both events occur is the product of their individual probabilities: \( P(A \cap B) = P(A) \times P(B) \). This rule extends to any number of independent events, so for three independent events \(A\), \(B\), and \(C\), \( P(A \cap B \cap C) = P(A) \times P(B) \times P(C) \).

For example, suppose a weather model reports a probability of 0.3 that it rains tomorrow and, independently, a probability of 0.5 that a particular stock rises tomorrow. Because these two events have no causal connection, the probability that both occur is \( 0.3 \times 0.5 = 0.15 \). Notice that the multiplication rule only applies once independence has been confirmed, either from the structure of the problem or by checking that conditioning on one event leaves the other's probability unchanged.

Many exam questions ask a student to decide whether two events are independent rather than simply stating it. The standard test is to compare \( P(A \cap B) \) with \( P(A) \times P(B) \): if the two quantities are equal, the events are independent; if not, they are dependent. For instance, if \( P(A) = 0.4 \), \( P(B) = 0.25 \), and a table or experiment shows \( P(A \cap B) = 0.1 \), then since \( 0.4 \times 0.25 = 0.1 \), the events are confirmed independent.

If instead \( P(A \cap B) \) had been reported as 0.2, the events would be dependent, since \( 0.2 \ne 0.1 \). This algebraic check is more reliable than intuition, because some relationships that appear connected in context turn out to satisfy the independence condition exactly, while others that seem unrelated do not.

When a problem involves repeated independent trials, a tree diagram organizes every possible sequence of outcomes and makes the multiplication rule visual. Each branch is labeled with its probability, and the probability of a full path from root to leaf is the product of the probabilities along that path.

Tree diagrams are especially useful when probabilities are not equal on each branch, since they keep the multiplication organized across several stages without requiring the student to track every combination mentally.

A factory machine has a 0.02 probability of producing a defective part on any given run, and successive runs are independent of one another. What is the probability that exactly the first two runs out of three are defective, while the third is not?

Since the runs are independent, the probability of this exact sequence is \( 0.02 \times 0.02 \times 0.98 \), which equals \( 0.000392 \). A common exam variation asks for the probability of at least one defective part across three runs, which is more efficiently found using the complement rule: \( 1 - (0.98)^3 \), since computing the complement of "no defects at all" avoids summing several overlapping cases.

Working through problems like this reinforces two habits that carry into more advanced statistics: confirming independence before multiplying, and choosing between direct calculation and the complement rule depending on which path involves fewer cases.

Independent Events: Two events where the occurrence of one does not change the probability of the other, so \( P(A \mid B) = P(A) \).

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

Multiplication Rule: For independent events, \( P(A \cap B) = P(A) \times P(B) \).

Conditional Probability: The probability of one event given that another has already occurred, written \( P(A \mid B) \).

Complement Rule: The probability that an event does not occur, calculated as \( 1 - P(A) \), often used to simplify "at least one" problems.

Related lessons

• Individual Event Idea Relationships