What are dependent events in probability?

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Dependent events in probability are events where the occurrence or outcome of one event affects the likelihood of the occurrence of another event. In other words, if two events are dependent, knowing that one event has occurred changes the probability of the other event happening.

Key points include:

1. Definition: For two events A and B, they are dependent if the probability of A given that B has occurred, denoted P(A | B), is different from the probability of A occurring on its own, P(A). That is, A and B are dependent if P(A | B) ≠ P(A).

2. Formula: When events are dependent, the probability of both events happening together (i.e., the joint probability) is given by:
  P(A and B) = P(B) × P(A | B)
This is in contrast to independent events, where the joint probability is simply P(A) × P(B).

3. Example:
  Imagine drawing two cards sequentially from a deck without replacement. Let:
  A = drawing an Ace as the first card,
  B = drawing an Ace as the second card.
  When the first card is drawn and an Ace is observed, the deck now contains one less Ace and one less card overall. Therefore, the probability for event B (drawing an Ace on the second draw) is altered by what happened in event A, making these events dependent.

Understanding dependent events is essential in probability theory as it helps determine correct probabilities in scenarios where outcomes influence one another.
Dependent events in probability are events where the outcome of one affects the outcome of another. For example, drawing two cards from a deck without replacement. If the first card drawn is an Ace, it changes the chances of drawing an Ace on the second draw, making these events dependent.

What are dependent events in probability?

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