To understand independent and dependent events, let's look at some simple examples:

Independent Events:

Imagine flipping a coin and rolling a die.

When you flip the coin, it doesn't change what happens when you roll the die.

So, if you want to find out the chances of getting heads on the coin and a 4 on the die, you can calculate it like this:

1. The chance of heads (P(A)) is 1 out of 2 (or 1/2).
2. The chance of rolling a 4 (P(B)) is 1 out of 6 (or 1/6).

To find the total chance (P(A and B)), you multiply these two together:

$P(A \text{ and } B) = P(A) \times P(B) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$

So, the chance of getting heads and rolling a 4 is 1 out of 12.

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Dependent Events:

Now, think about drawing cards from a deck without putting any cards back.

If you pick an ace first, there are fewer cards to choose from the next time.

This means the chances of drawing another ace change.

Here's how you can calculate it:

1. The chance of drawing an ace first (P(A)) is 4 out of 52 cards (or 4/52).
2. If you already drew an ace, now there are only 3 aces left and 51 cards total. So, the chance of drawing an ace next (P(B | A)) is now 3 out of 51 (or 3/51).

To find the total chance (P(A and B)), you multiply these together:

$P(A \text{ and } B) = P(A) \times P(B | A) = \frac{4}{52} \times \frac{3}{51}$

This helps show how these two types of events are different!