In probability, the formula for dependent events accounts for how the occurrence of one event affects the likelihood of another. This is different from independent events, where events have no impact on each other.

For two dependent events $A$ and $B$, the probability of both events occurring (often noted as $P(A \cap B)$) can be found using the conditional probability:

$P(A \cap B) = P(A) \cdot P(B|A)$

where:

• $P(A)$ is the probability of event $A$ occurring.
• $P(B|A)$ is the probability of event $B$ occurring given that event $A$ has already occurred.

This formula highlights that for dependent events, we adjust the probability of the second event based on the outcome of the first event.

Example

Suppose you have a deck of cards and draw two cards without replacement:

1. Let $A$ be the event that the first card drawn is an Ace.
2. Let $B$ be the event that the second card drawn is an Ace.

Since drawing the first card affects the likelihood of drawing an Ace the second time (by reducing the deck size and number of Aces), these events are dependent.

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Would you like a detailed example or have further questions?

Here are related questions to deepen your understanding:

1. What is the difference between dependent and independent events in probability?
2. How does the conditional probability formula relate to dependent events?
3. Can you calculate dependent probabilities for more than two events?
4. What are real-life examples of dependent events?
5. How is Bayes’ theorem related to dependent events?

Tip: When calculating probabilities for dependent events, always consider how the first event impacts the conditions or available outcomes for the second event.