Coin Toss Probability, Worked Out Step by Step

Coin flips are the classic binomial example for a reason — everyone already understands the setup, so it's the easiest way to actually see the math working.

The question

If you flip a fair coin 10 times, what's the probability of getting exactly 6 heads?

Setting it up

• n = 10 (number of flips)
• p = 0.5 (probability of heads on one flip)
• k = 6 (the number of heads we're asking about)

Plugging into the formula

P(X = 6) = C(10, 6) · 0.5⁶ · 0.5⁴

C(10, 6) is "10 choose 6," which works out to 210 — that's the number of different orders you could land exactly 6 heads and 4 tails in across 10 flips. Multiply that by 0.5⁶ (the heads part) and 0.5⁴ (the tails part), and you land on approximately 0.205, or about 20.5%.

Not exactly what most people guess. A lot of people assume 6 out of 10 should be pretty likely since it's "close to half," and it is one of the more likely individual outcomes — but any single specific number of heads out of 10 tops out around 24-25% at most, which happens right at 5 heads.

What about "at least 6 heads"?

This is where people usually go wrong — they calculate P(X = 6) and stop, when the question actually asked for "at least." P(X ≥ 6) means adding up the probabilities of getting 6, 7, 8, 9, and 10 heads. That total comes out to roughly 0.377, or about 37.7% — nearly double the "exactly 6" answer. Big difference, and it's exactly why picking the right mode on the calculator matters.

Try it yourself

Punch n = 10, p = 0.5, k = 6 into the calculator and switch between "exactly," "at least," and "at most" to see how much the answer swings. It's honestly the fastest way to get an intuition for cumulative probability.

Scaling it up

The same logic works for any number of flips. Flip 100 times and ask for exactly 50 heads, and you'll notice the probability of that one exact number actually drops — even though 50 is still the most likely single outcome, there are just so many more possible outcomes (0 through 100) competing for probability. This is the discrete nature of binomial distribution showing up directly — see the beginner's guide for more on that.