When learning about basic probability, it's really helpful to understand how addition and multiplication rules work, especially with independent events. Let's go through a couple of easy examples:

Addition Rule:

Imagine you have a bag of marbles. There are 3 red ones and 2 blue ones. If you want to find out the chance of picking either a red marble or a blue marble, you can use the addition rule.

1. First, let's find the chance of picking a red marble:

   • ( P(\text{Red}) = \frac{3}{5} ) (This means there are 3 chances to get a red marble out of 5 total marbles.)

2. Next, we find the chance of picking a blue marble:

   • ( P(\text{Blue}) = \frac{2}{5} ) (This means there are 2 chances to get a blue marble out of 5 total marbles.)

Now, to find the total chance of picking either a red or a blue marble, we add these two chances together: $P(\text{Red or Blue}) = P(\text{Red}) + P(\text{Blue}) = \frac{3}{5} + \frac{2}{5} = 1$

This means there’s a 100% chance you’ll pick either a red or a blue marble since those are the only two colors in the bag.

Multiplication Rule:

Now, let’s think about flipping a coin and rolling a die. The chance of the coin landing on heads ($H$) is ( P(H) = \frac{1}{2} ) (which is 50% chance). The chance of rolling a 4 on the die is ( P(4) = \frac{1}{6} ) (which is about 16.67% chance). Since flipping the coin and rolling the die don’t affect each other, we can multiply their chances to find out how likely it is to get both results:

$P(H \text{ and } 4) = P(H) \times P(4) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$

So, there’s a 1 in 12 chance (or about 8.33%) of getting heads when you flip the coin and rolling a 4 at the same time.

These examples show how the addition and multiplication rules help us figure out probabilities in different situations much more easily.