Real-World Examples of the Addition and Multiplication Rules

Understanding the Rules
In probability, we have two important rules: the Addition Rule and the Multiplication Rule.

• The Addition Rule helps us find the chance of one event or another happening.
• The Multiplication Rule helps us find the chance of two events happening at the same time.

Addition Rule Example:
Let’s say you want to find the chance of drawing a certain card from a deck of 52 cards.

If you want to know the chance of getting either an Ace or a King, you can use the Addition Rule.

The chance of drawing an Ace, called $P(A)$, is:

$$P(A) = \frac{4}{52} = \frac{1}{13}$$

The chance of drawing a King, called $P(K)$, is also:

$$P(K) = \frac{4}{52} = \frac{1}{13}$$

Since you can’t draw a card that is both an Ace and a King, we can add the two probabilities together:

$$P(A \cup K) = P(A) + P(K) = \frac{1}{13} + \frac{1}{13} = \frac{2}{13}$$

So, the chance of drawing either an Ace or a King is about 0.154, or 15.4%.

Multiplication Rule Example:
Now let’s look at a situation where you flip two coins. If you want the chance of getting Heads on both coins, you can use the Multiplication Rule.

The chance of the first coin landing on Heads, called $P(H_1)$, is:

$$P(H_1) = \frac{1}{2}$$

The chance for the second coin landing on Heads, called $P(H_2)$, is also:

$$P(H_2) = \frac{1}{2}$$

Since the flips do not affect each other, we multiply the two probabilities:

$$P(H_1 \cap H_2) = P(H_1) \times P(H_2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$

This means there is a 25% chance of flipping two Heads.

More Context:
Now, let’s think of a marketing example. Suppose the chance of a customer clicking on an ad is 0.3, and the chance of that customer buying something after clicking is 0.4.

To find the chance that a customer clicks the ad and then makes a purchase, we use the Multiplication Rule:

$$P(Click \cap Purchase) = P(Click) \times P(Purchase | Click) = 0.3 \times 0.4 = 0.12$$

So there is a 12% chance that a customer clicks the ad and also makes a purchase.

These examples show how the Addition and Multiplication Rules can be used in real life. They help us make better choices by understanding different situations.