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What does "not all same" mean?
"Not all same" means getting a mixture of heads and tails — at least one of each. In other words, your flips aren't all identical.
What is "all same"? (The complement)
"All same" means either all heads OR all tails. These are the only two ways to get uniform results. Everything else is "mixed".
All Heads
All Tails
Why use complement here?
"Mixed results" has many possibilities (HT, TH, HTH, THH, TTH, etc.). But "all same" has only TWO possibilities (all H or all T) — much easier to count!
The Formula:
Flip a fair coin twice. What's the probability of getting a mix (not all heads, not all tails)?
Let's list all 4 possible outcomes:
Green = mixed (what we want) | Red = all same (complement)
Calculate P(all same)
"All same" can happen two ways: HH or TT. Each has probability ¼:
Apply complement rule
"Mixed" is everything except "all same":
Result:
✓ Verification: 2 out of 4 outcomes (HT, TH) are mixed = 2/4 = ½ ✓
Flip a fair coin three times. What's the probability of getting a mixture of heads and tails?
Why complement shines: With 3 flips, there are 2³ = 8 outcomes. Only 2 are "all same" (HHH and TTT), while 6 are "mixed". It's easier to count 2 than 6!
Calculate P(all same)
P(HHH) = (½)³ = ⅛ and P(TTT) = (½)³ = ⅛
Apply complement
Result:
| Flips | P(all same) | P(mixed) |
|---|---|---|
| 2 | ||
| 3 | ||
| 4 | ||
| 5 |
Note: For n flips, P(all same) = 2 × (1/2)^n = (1/2)^(n-1)
As the number of flips increases, getting all the same result becomes increasingly unlikely, making mixed results almost certain for large n.