Independent versus dependent events

The key question is not whether two events happen close together. It is whether learning that the first happened changes the probability of the second. That determines whether you can multiply the original probabilities or must update the second probability first.

Independent events

Events A and B are independent when P(B given A) equals P(B). In words, A happening does not change the chance of B. For independent events, the probability that both occur is P(A and B) = P(A) × P(B). A fair coin flip followed by a separate fair die roll is a standard example: the coin result does not change the die’s probabilities.

Dependent events

Events are dependent when the first changes the probability of the second. For a sequence, P(A and B) = P(A) × P(B given A). The second factor is the probability after the first event has occurred, not necessarily the original probability.

For example, a bag contains 3 red and 2 blue counters. Draw one counter and keep it out, then draw another. The probability of red on the first draw is 3/5. If the first counter was red, 2 red remain among 4 counters, so the chance of red on the second draw is 2/4. The chance of two reds is 3/5 × 2/4 = 3/10. The first draw changes the contents, so the events are dependent.

Replacement often decides the setup

Test independence with conditional probability

If P(B given A) equals P(B), the events are independent, provided the probabilities are defined. If the values differ, they are dependent. The multiplication rule gives the same test: A and B are independent when P(A and B) = P(A)P(B). Do not assume events are independent just because the problem does not explicitly say “without replacement”; use the experiment’s wording and conditions.

Common errors

• Multiplying the original probabilities in a no-replacement problem without updating the counts.
• Assuming two events are dependent merely because they occur in sequence.
• Confusing “A and B” (both occur) with “A or B” (at least one occurs).
• Treating mutually exclusive events as independent. If one of two mutually exclusive events occurs, the other cannot occur; unless one has probability zero, that change makes them dependent.
• Forgetting that replacement restores the sample space only when the item is returned and the sample is mixed as the problem specifies.

A quick setup routine

1. Identify the first event and the second event.
2. Ask whether the first event changes the sample space or the probability of the second.
3. If nothing changes, multiply the original probabilities.
4. If something changes, calculate the second probability using the new conditions, then multiply.
5. Keep fractions exact until the final step and check that the joint probability is no larger than either individual probability.

A short phrase often identifies the model: “with replacement” usually preserves the probability, while “without replacement” usually changes it. Always confirm that the story supports that interpretation before calculating.

Test whether one event changes the other probability

Two events are independent when knowing that one occurred does not change the probability of the other. In notation, P(A given B) = P(A), and for an intersection P(A and B) = P(A)P(B). Independence is a relationship defined by probabilities, not by physical separation or by whether events happen at the same time.

A fair coin tossed twice has independent outcomes: the first toss does not change the chance of heads on the second, which remains 1/2. The probability of heads twice is 1/2 × 1/2 = 1/4. By contrast, drawing two cards from a deck without replacement creates dependence because the first card changes the remaining deck and the probability on the second draw.

Account for replacement and changing conditions

With replacement, the selected object is returned and the composition of the population is restored, so repeated draws are often modeled as independent. Without replacement, probabilities change after each draw. If a bag has 3 red and 2 blue tokens, the probability of red on the first draw is 3/5. Without replacement, if the first token is red, the next red probability is 2/4, not 3/5.

Real-world events may be dependent even when a problem gives limited detail. Rain can affect traffic delays; studying time and exam score can be related through preparation or other factors. Do not assume independence merely because the events are listed separately. Use the stated model or compare the conditional and unconditional probabilities if the data allow it.

Distinguish independence from mutual exclusivity

Mutually exclusive events cannot both occur in the same trial. Independent events do not affect each other’s probability. Events with positive probabilities that are mutually exclusive are dependent: if A occurs, B is known not to occur, so P(B given A) = 0 rather than P(B). For example, one coin toss cannot be both heads and tails, but results of separate tosses can be independent.

For a pair of events, test the correct relationship: use addition with an overlap term for “A or B” and multiplication for “A and B.” Independence permits replacing P(A and B) with P(A)P(B), but only when independence is justified. Mutually exclusive events use P(A and B) = 0. These rules answer different questions.

• Ask whether learning one event occurred changes the probability of the other.
• Use multiplication for independent intersections: P(A and B) = P(A)P(B).
• Without replacement, recalculate after the population changes.
• Do not confuse independent events with mutually exclusive events.
• Use conditional probabilities when the problem supplies them.

Use the experiment to decide whether probabilities change

Independence is a property of the event process, not merely whether two event names differ. Drawing a card, replacing it, and shuffling before a second draw leaves the deck composition unchanged; the probability of a second heart remains 13/52. Without replacement, a first heart leaves 12 hearts among 51 cards, so the probability changes and the draws are dependent. The product rule still applies in both cases, but for dependent events use the conditional probability after the first outcome.

For events A and B, independence means P(B given A) = P(B), equivalently P(A and B) = P(A)P(B). One useful check is to compare the conditional probability with the original probability. If they differ, the events are dependent. A two-way table can help: calculate the proportion with B among the group that has A and compare it with the overall proportion with B. Do not assume events are independent because they occur at different times or involve different people; the wording must support that assumption.