Coin Flip Probability Calculator

Calculate the probability of getting a specific number of heads or tails in a series of coin flips using binomial probability.

Calculate Your Coin Flip Probability Calculator

Number of Coin Flips

Enter a value between 1 and 1000

Probability Type

Choose whether to calculate the probability of getting exactly, at least, or at most the specified number of heads

Number of Heads

Enter a value between 0 and the number of flips (10)

Coin Type

Fair Coin (50%)

Biased Coin

Probability of Heads (0-1)

For a fair coin, use 0.5. For a biased coin, enter a value between 0 and 1

Understanding Coin Flip Probability

Coin flip probability is a fundamental concept in probability theory that studies the likelihood of specific outcomes when flipping a coin one or more times. While a single coin flip is simple (with a 50% chance of heads and 50% chance of tails for a fair coin), calculating the probability of specific patterns across multiple flips becomes more complex.

These probabilities are governed by the binomial probability distribution, which is applicable whenever we have a fixed number of independent trials, each with the same probability of success.

The Binomial Probability Formula

For coin flips, the binomial probability formula calculates the probability of getting exactly k heads in n flips:

P(X = k) = (n choose k) × p^k × (1-p)^(n-k)

Where:

n = number of flips
k = number of heads
p = probability of heads on a single flip (0.5 for a fair coin)
(n choose k) = the binomial coefficient, calculated as n! / (k! × (n-k)!)

Types of Probability Calculations

Exactly k Heads

The probability of getting exactly k heads in n flips, using the binomial formula directly.

P(X = k) = (n choose k) × p^k × (1-p)^(n-k)

At Least k Heads

The probability of getting k or more heads in n flips, calculated by summing individual probabilities.

P(X ≥ k) = Σ(i=k to n) P(X = i)

At Most k Heads

The probability of getting k or fewer heads in n flips, also calculated by summing individual probabilities.

P(X ≤ k) = Σ(i=0 to k) P(X = i)

Common Coin Flip Probability Scenarios

Scenario	Formula	Example	Probability
Getting all heads	p^n	10 heads in 10 flips	0.098% (1/1024)
Getting all tails	(1-p)^n	0 heads in 10 flips	0.098% (1/1024)
Equal number of heads and tails	(n choose n/2) × p^(n/2) × (1-p)^(n/2)	5 heads in 10 flips	24.61% (245/1024)
More heads than tails	Σ(i=n/2+1 to n) P(X = i)	6+ heads in 10 flips	37.70%
At least one head	1 - (1-p)^n	1+ heads in 10 flips	99.90%

Fair vs. Biased Coins

While most probability examples assume a fair coin (p = 0.5), real-world coins might have slight biases:

Fair Coin: Equal probability of heads and tails (p = 0.5)
Biased Coin: Unequal probability (e.g., p = 0.6 means 60% chance of heads)

Physical coins typically have small biases due to manufacturing imperfections, with studies suggesting that a coin has approximately a 51% chance of landing on the same face it started on. Our calculator allows you to account for biased coins by adjusting the probability of heads.

Applications of Coin Flip Probability

Understanding coin flip probability has numerous practical applications:

Educational Tool: Teaching concepts of probability, expected value, and variance
Gaming & Gambling: Analyzing odds in games of chance
Statistical Testing: Coin flips represent Bernoulli trials, a foundation for many statistical tests
Computer Science: Randomized algorithms and probabilistic data structures
Cryptography: Coin flipping protocols for secure multi-party computation
Decision Theory: Modeling choices with uncertain outcomes

Related Calculators

Coin Flipper Calculator

Flip virtual coins to simulate probability experiments and make random binary decisions

Probability Calculator

Calculate the probability of various events occurring based on given parameters and distributions

Coin Toss Streak Calculator

Calculate probabilities of consecutive heads or tails in a series of coin tosses

Implied Probability Calculator

Convert betting odds to implied probability percentages

Frequently Asked Questions

For a fair coin, the probability of getting all heads in n consecutive flips is (1/2)^n. For example, the probability of getting all heads in 5 flips is (1/2)^5 = 1/32 ≈ 3.125%. For a biased coin with probability p of heads, the formula is p^n.

Use the binomial probability formula: P(X=3) = (5 choose 3) × (0.5)^3 × (0.5)^2. The binomial coefficient (5 choose 3) equals 10, so the calculation is 10 × 0.125 × 0.25 = 0.3125 or 31.25%. This means there's a 31.25% chance of getting exactly 3 heads in 5 flips of a fair coin.

While getting exactly 50% heads becomes increasingly unlikely as the number of flips increases, the probability of getting close to 50% actually increases. This is due to the Law of Large Numbers. For example, with 100 flips, the probability of exactly 50 heads is only about 8%, but the probability of 45-55 heads is about 73%. The distribution peaks more sharply around the expectation with more flips.

'At least k heads' means k or more heads (calculating P(X ≥ k)), while 'at most k heads' means k or fewer heads (calculating P(X ≤ k)). For example, 'at least 3 heads in 5 flips' includes the possibilities of getting 3, 4, or 5 heads. 'At most 3 heads in 5 flips' includes the possibilities of getting 0, 1, 2, or 3 heads.

A biased coin has unequal probabilities of heads and tails. If p is the probability of heads (e.g., p = 0.6), then all binomial probability calculations use this p value instead of 0.5. For example, the probability of getting exactly 3 heads in 5 flips with a coin that has p = 0.6 would be (5 choose 3) × (0.6)^3 × (0.4)^2 = 10 × 0.216 × 0.16 = 0.3456 or 34.56%.

For very large numbers of flips (typically thousands or more), direct calculation using the binomial formula can lead to computational issues due to extremely large factorials and very small powers. For large n, statistical approximations like the normal approximation to the binomial distribution are often used. Our calculator limits calculations to 1000 flips to ensure accuracy and reasonable computation time.

For a fair coin, the expected number of heads in n flips is n/2. For a biased coin with probability p of heads, the expected number is n×p. For example, if you flip a fair coin 100 times, you can expect to get about 50 heads on average. This is the mean of the binomial distribution. The standard deviation is √(n×p×(1-p)), which for a fair coin simplifies to √(n/4).

Share This Calculator

Found this calculator helpful? Share it with your friends and colleagues!