Key Differences Between Independent and Dependent Combined Events

1. What They Mean:

   • Independent Events: These are events where what happens in one does not change what happens in the other. For example, flipping a coin and rolling a die at the same time.
   • Dependent Events: In these events, what happens in one does affect what happens in the other. For example, if you draw cards from a deck without putting the first one back.

2. How to Calculate Probability:

   • Independent Events: To find the chance of both happening, we multiply their chances: $P(A \text{ and } B) = P(A) \times P(B)$
   • Dependent Events: To find the chance of both happening, we also multiply, but we take into account how the first affects the second: $P(A \text{ and } B) = P(A) \times P(B | A)$

3. Examples:

   • Independent: If you want to know the chance of rolling a 3 on a die (which is 1 out of 6) and flipping tails on a coin (which is 1 out of 2), you calculate it like this: $P = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
   • Dependent: If you want to know the chance of drawing two aces from a deck of cards without putting the first one back, it looks like this: $P = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} \approx 0.0045$