To understand how addition and multiplication rules work together to calculate compound probabilities, it's best to keep things simple.

Addition Rule

The addition rule helps us find the chance of one event happening or another.

For example, let’s say you are rolling a die and want to know what the chance is of rolling a 3 or a 4.

To find this, you add the chances of each number. Since a die has six sides, the chance of rolling a 3 is $\frac{1}{6}$, and the chance of rolling a 4 is the same.

So, using the addition rule, it looks like this:

$P(3 \text{ or } 4) = P(3) + P(4) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

This rule is important when the two events can't happen at the same time. We call those mutually exclusive events.

Multiplication Rule

Now, the multiplication rule is used when you want to find the chance of two independent events happening together.

Imagine you toss a coin and roll a die.

The chance of the coin showing heads ($P(H) = \frac{1}{2}$) and then rolling a 3 ($P(3) = \frac{1}{6}$) is found by multiplying the two chances:

$P(H \text{ and } 3) = P(H) \times P(3) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$

Interaction of Rules

When we look at compound probabilities, we often use both rules together.

For example, if you want the chance of rolling a 3 or 4 and getting heads on a coin toss, you start by using the addition rule for the die:

$P(3 \text{ or } 4) = \frac{1}{3}$

Then, you multiply by the chance of the coin toss:

$P(H) = \frac{1}{2}$

Putting it all together gives:

$P((3 \text{ or } 4) \text{ and } H) = P(3 \text{ or } 4) \times P(H) = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

In short, the addition and multiplication rules work together to help us understand probabilities in a clear way. This is true whether we are looking at events that can happen at the same time or events that happen independently.