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Dependent Events

Dependent events are events where the outcome of one event changes the probability of the other event happening. For example, if you draw a card from a deck and don't put it back, the probabilities for the second draw change because there are fewer cards left.

Two events AA and BB are dependent if the occurrence of one event affects the probability of the other. Formally, events are dependent when P(B∣A)≠P(B)P(B \mid A) \neq P(B), meaning the probability of BB given that AA has occurred is different from the probability of BB on its own. The probability that both dependent events occur is found by multiplying the probability of the first event by the conditional probability of the second event given the first.

Key Formula

P(A and B)=P(A)×P(B∣A)P(A \text{ and } B) = P(A) \times P(B \mid A)
Where:
  • P(A and B)P(A \text{ and } B) = the probability that both events A and B occur
  • P(A)P(A) = the probability that event A occurs
  • P(B∣A)P(B \mid A) = the probability that event B occurs given that event A has already occurred

Worked Example

Problem: A bag contains 5 red marbles and 3 blue marbles. You draw one marble without replacing it, then draw a second marble. What is the probability that both marbles are red?
Step 1: Find the probability that the first marble is red. There are 5 red marbles out of 8 total.
P(1st red)=58P(\text{1st red}) = \dfrac{5}{8}
Step 2: After removing one red marble, the bag now has 4 red and 3 blue marbles — 7 marbles total. Find the probability the second marble is also red.
P(2nd red∣1st red)=47P(\text{2nd red} \mid \text{1st red}) = \dfrac{4}{7}
Step 3: Multiply the two probabilities together using the dependent events formula.
P(both red)=58×47=2056=514P(\text{both red}) = \dfrac{5}{8} \times \dfrac{4}{7} = \dfrac{20}{56} = \dfrac{5}{14}
Answer: The probability of drawing two red marbles without replacement is 514\dfrac{5}{14}, which is approximately 0.357 or about 35.7%.

Visualization

Why It Matters

Dependent events show up whenever a situation changes as actions are taken. Card games, lottery draws, and quality-control inspections all involve dependent events because items are removed from the pool as selections happen. Understanding dependence helps you avoid overestimating or underestimating the likelihood of combined outcomes.

Common Mistakes

Mistake: Using the same probability for the second event as the first (treating dependent events as independent).
Correction: When events are dependent, the sample size or conditions change after the first event. You must update the probability for the second event to reflect what has already happened — for instance, reducing the total number of items by one after drawing without replacement.
Mistake: Forgetting to adjust both the numerator and the denominator after the first event.
Correction: If you draw a red marble from a bag, the total number of marbles decreases by one AND the number of red marbles decreases by one. Make sure you update both counts.

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