Addition Rule

What is the Addition Rule of Probability?

Imagine asking your classmates, "How did you get to school today?". Then you want to know, what's the probability if we randomly pick one student, that they came by bike OR motorcycle?

Well, the word "OR" is super important in the world of probability. The Addition Rule is how we calculate the combined probability of two (or more) events using the word "OR". It means we want to know the probability that event A happens, or event B happens, or even both happen (if that's possible).

There are two important situations when we talk about "OR":

1. The events cannot happen at the same time (These are called Mutually Exclusive events).
2. The events can happen at the same time (These are called Not Mutually Exclusive events).

Let's break them down!

Mutually Exclusive Events

Mutually Exclusive events (or the fancy names Disjoint events) mean two events that cannot possibly happen at the same time in a single trial.

If event A happens, then event B definitely doesn't happen, and vice versa.

Simple Examples:

1. You flip a coin. The result is either "Heads" or "Tails". It's impossible to get both "Heads" and "Tails" at the same time on one coin, right?
2. Zain rode a motorcycle to school today OR Zain rode a bike to school today. (Usually, a person uses only one mode of transport at a time, right? So these are mutually exclusive).
3. Rolling two dice: The event "the sum is 7" OR the event "getting doubles (same number on both dice)".
   • Sum is 7: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$
   • Doubles: $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$
   • Look! There are no pairs that are the same in both groups. So, these two events are mutually exclusive!

Formula for Mutually Exclusive Events

If events A and B are mutually exclusive, calculating the probability of "A OR B" is super easy – just add their individual probabilities:

Calculation Example:

Using the two-dice roll from before:

• Probability of sum being 7 ($P(S=7)$) = There are 6 pairs out of 36 total = $6/36$
• Probability of getting doubles ($P(\text{Doubles})$) = There are 6 pairs out of 36 total = $6/36$

So, the probability of getting a sum of 7 OR doubles is:

Not Mutually Exclusive Events

Not Mutually Exclusive events mean two events that CAN happen at the same time in a single trial. There's a possibility that both events occur together.

Simple Examples:

1. Drawing one card from a standard deck. The event "getting an Ace" OR the event "getting a Heart ($\heartsuit$)". Can these happen together? Sure! The Ace of Hearts ($A\heartsuit$).
2. Rolling two dice: The event "the sum is 8" OR the event "getting doubles".
   • Sum is 8: $(2,6), (3,5), (4,4), (5,3), (6,2)$
   • Doubles: $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$
   • Hey, look! There's an overlap, right? The pair $(4,4)$. Because there's an overlap, these two events are not mutually exclusive.

Can we just add them up?

If we just add $P(A) + P(B)$ for non-mutually exclusive events, we'll count something twice! That something is the part where A and B happen together (their intersection).

Like in the sum 8 OR doubles example:

• $P(S=8)$ = There are 5 pairs = $5/36$
• $P(\text{Doubles})$ = There are 6 pairs = $6/36$

If we just add them: $5/36 + 6/36 = 11/36$. The pair $(4,4)$ was counted in the "Sum is 8" group AND counted again in the "Doubles" group. So it was counted twice!

Formula for Non-Mutually Exclusive Events

For events A and B that are not mutually exclusive, the way to calculate the probability of "A OR B" is:

The $P(A \text{ and } B)$ part is the probability that events A and B happen at the same time (the intersection). We subtract this so we don't count it twice.

Calculation Example:

Let's continue the sum 8 OR doubles example:

• $P(S=8) = 5/36$
• $P(\text{Doubles}) = 6/36$
• The probability of the event "sum is 8 AND doubles" ($P(S=8 \text{ and Doubles})$) is just one outcome: the pair (4,4). So its probability is $1/36$.

Therefore, the probability of getting a sum of 8 OR doubles is:

Important Note About the Word "OR"!

In mathematics, "A or B" means it could be A only, B only, or BOTH A and B (if possible). This is different from everyday "or" which sometimes means "pick one or the other". Remember, "or" in math is inclusive!

When to Use Which Formula?

1. First, check: Can events A and B happen at the same time?
2. If they CANNOT happen together: Use the simple formula $P(A \text{ or } B) = P(A) + P(B)$.
3. If they CAN happen together: Use the full formula $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.