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":
The events cannot happen at the same time (These are called events).
Mutually Exclusive
The events can happen at the same time (These are called Not 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:
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?
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).
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!
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:
Drawing one card from a standard deck. The event "getting an Ace" OR the event "getting a Heart (♡)". Can these happen together? Sure! The Ace of Hearts (A♡).
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, so the probability is 5/36
P(Doubles) is 6 pairs, namely 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!
For events A and B that are not mutually exclusive, the way to calculate the probability of "A OR B" is:
P(A orB)=P(A)+P(B)−P(A andB)
The P(A andB) 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(Doubles)=6/36
The probability of the event "sum is 8 AND doubles" (P(S=8 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:
P(S=8 or Doubles)=P(S=8)+P(Doubles)−P(S=8 and Doubles)
=365+366−361=3610=185
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!
