Coin Flip Probability
The Science of Chance
Flipping a coin is the universal symbol of randomness. It is a 50/50 shot. But when you start flipping a coin multiple times, the math gets complicated quickly.
This calculator uses the Binomial Distribution formula to predict outcomes. It answers complex questions like: "If I flip a coin 10 times, what are the odds that I get AT LEAST 7 heads?"
The Binomial Formula
To calculate the probability ($P$) of getting exactly $k$ heads in $n$ flips, statisticians use this equation:
- n: Total number of flips.
- k: Number of heads desired.
- p: Probability of heads (0.5 for a fair coin).
- nCk: "n Choose k" (Combination formula).
The Gambler's Fallacy
This is the biggest mistake people make with probability. Imagine you have flipped a coin 5 times, and it landed on Heads every single time. What are the odds that the next flip will be Tails?
Most people feel like it is "Due" to be Tails. They think the universe needs to balance it out.
The coin has no memory. It does not know what happened in the past. Every single flip is an Independent Event. The previous 5 heads have zero influence on the 6th flip.
The Law of Large Numbers
If the coin has no memory, why do we say it is 50/50? Because of the Law of Large Numbers.
If you flip a coin 10 times, getting 70% heads is actually quite common (17% chance). It is a small sample size.
However, if you flip a coin 1,000,000 times, it is statistically almost impossible to get 70% heads. As the number of trials increases, the actual ratio will get closer and closer to the theoretical 50/50 split.
Probability Chart (10 Flips)
What are the odds if you flip a coin 10 times?
| Exact Outcome | Probability | Percent |
|---|---|---|
| 5 Heads | 0.2461 | 24.6% |
| 6 Heads | 0.2051 | 20.5% |
| 0 Heads | 0.0010 | 0.1% (Very rare!) |
| 10 Heads | 0.0010 | 0.1% (Very rare!) |