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Plain English Definition:
The complement of an event is simply "everything else that could happen."If you don't get what you want, you get the complement instead.
Think of it this way:
The Key Insight:
All probabilities must add up to 1 (or 100%). This is because something has to happen!
Since an event either happens or it doesn't:
Rearranging:
Example with a Die:
If P(rolling a 6) = 1/6, then the remaining probability (5/6) must go to "NOT rolling a 6". Think of probability like a pie: if 6 takes 1/6 of the pie, everything else gets the remaining 5/6.
What is the probability of rolling a fair die and NOT getting a 6?
These are the outcomes we want (everything except 6):
First, find P(rolling a 6)
A die has 6 faces, each equally likely. Rolling a 6 is just 1 out of 6 possibilities.
Subtract from 1 to get the complement
Since probabilities add to 1, whatever is left after "rolling a 6" goes to "NOT rolling a 6".
Verify by counting directly
Count the favorable outcomes: 1, 2, 3, 4, 5 = 5 outcomes out of 6 total.
Final Answer:
There's about an 83% chance of NOT rolling a 6.
Use complement when "NOT happening" is easier to count than "happening."
✓ Good for Complement:
"At least one 6 in 10 rolls"
Easier to find "no 6s at all" than list all ways to get 1, 2, 3... or 10 sixes
Could use either:
"NOT rolling a 6"
Can count 5 outcomes directly, or do 1 - 1/6
For any single number on a fair die:
Rolling that number
NOT rolling that number