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What does "at least one" mean?
"At least one head" means one or more heads. It includes:
Basically, anything EXCEPT getting no heads at all (all tails).
Why is complement the smart approach?
The hard way: Add up P(exactly 1 head) + P(exactly 2 heads) + P(exactly 3 heads)...
The easy way: Find P(no heads) and subtract from 1. Since "at least one head" is the complement of "no heads", this gives us our answer in one step!
The Key Insight:
"At least one" = "NOT none" = 1 - P(none)
For coins: "At least one head" = 1 - P(all tails)
The Formula:
where n = number of coin flips
Flip a fair coin twice. What's the probability of getting at least one head?
Let's first list all possible outcomes (2 flips = 4 possibilities):
Green = at least one H (what we want) | Red = all T (complement)
Identify the complement: "all tails"
Looking at our outcomes, only ONE gives us "no heads": TT. This is much easier to count than the 3 outcomes with at least one head!
Calculate P(all tails)
Each flip has P(T) = ½. For both flips to be tails, we multiply (independent events):
Apply the complement rule
"At least one head" is everything EXCEPT "all tails":
Result:
✓ Verification: 3 out of 4 outcomes (HH, HT, TH) have at least one head = 3/4 ✓
Flip a fair coin three times. What's the probability of getting at least one head?
Why complement is even more valuable here: With 3 flips, there are 2³ = 8 possible outcomes. We'd need to count 7 of them for "at least one head", but only 1 for "all tails" (TTT).
P(all tails in three flips)
Multiply the probability of tails for each flip:
Apply complement
Result:
| Flips | P(all tails) | P(at least one H) |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 |