Probability of Exactly 3 Heads in 10 Coin Flips
A fair coin has two equally likely outcomes on each independent flip. This is a binomial setting: there is a fixed number of trials, each trial has two outcomes, and the probability of heads stays the same.
The question
What is the probability of getting exactly 3 heads when a fair coin is flipped 10 times?
These calculations use a probability model that matches the stated assumptions. If the events are not independent or the chance changes between trials, the binomial formula may not describe the real situation.
Step-by-step solution
- Set n = 10 flips, k = 3 heads, and p = 0.5 for heads. The probability of tails is also 0.5.
- There are C(10, 3) = 120 different orders in which 3 heads can occur among 10 flips.
- The probability of any one such sequence is (0.5)³ × (0.5)⁷ = (0.5)¹⁰.
- Multiply the number of sequences by the probability of each: C(10, 3)(0.5)³(0.5)⁷ = 120/1024 = 0.1171875.
Answer: The probability is 0.1171875, or about 11.72%. This is the chance of exactly 3 heads; it does not include outcomes with 2 or 4 heads.
Check the result with a calculator
Open the binomial calculator with n = 10, p = 0.5 and k = 3. You can change the inputs to compare a different number of trials, success probability or target count.
For other probability questions, use the binomial distribution calculator for exact, cumulative and range outcomes, or browse the full calculator directory.