Independent and dependent events are important ideas in probability. They help us understand how one event can relate to another.

Independent Events

Independent events are situations where one event does not affect another. This means that if event A happens, it doesn't change the chances of event B happening.

Example: Imagine tossing a coin and rolling a die.

• The chance of getting heads (Event A) is 1 out of 2, or 50%:
  P(A) = 1/2

• The chance of rolling a 4 (Event B) is 1 out of 6, or about 16.67%:
  P(B) = 1/6

To find the chance of both events happening together, we multiply their probabilities:
P(A and B) = P(A) × P(B) = 1/2 × 1/6 = 1/12

Dependent Events

Dependent events are different. In these cases, one event affects the chance of the other event happening. A good example is when you draw cards from a deck and do not put the first card back.

Example: Let’s say you draw two cards from a regular deck of 52 cards.

• The chance of drawing an Ace first (Event A) is 4 out of 52, or 1 out of 13:
  P(A) = 4/52 = 1/13

After drawing one Ace, the chance of drawing another Ace (Event B) changes. Now, there are only 3 Aces left and 51 cards total:
P(B|A) = 3/51

To find the chance of both events happening together, we multiply the probabilities again:
P(A and B) = P(A) × P(B|A) = 4/52 × 3/51 = 12/2652, which is about 1/221.

Summary

• Independent Events:

  • They do not affect each other.
  • To calculate: P(A and B) = P(A) × P(B).

• Dependent Events:

  • Their chances change based on what happened before.
  • To calculate: P(A and B) = P(A) × P(B|A).

Understanding these concepts helps us make sense of how events work together in probability!