Coin Toss Probability Calculator
Calculate the probability of getting a specific number of heads or tails in multiple coin flips using binomial distribution.
Coin Toss Probability Calculator
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Formula: P(X = k) = C(n,k) x p^k x (1-p)^(n-k)
Worked example — Exactly 7 heads: 11.72% probability | At least 7 heads: 17.19%
Formula
P(X = k) = C(n,k) x p^k x (1-p)^(n-k)
Where P(X = k) is the probability of exactly k heads, C(n,k) is the binomial coefficient (number of combinations), n is the number of flips, k is the desired number of heads, p is the probability of heads per flip (0.5 for a fair coin), and (1-p) is the probability of tails.
Worked Examples
Example 1: Probability of Exactly 7 Heads in 10 Flips
Problem:What is the probability of getting exactly 7 heads when flipping a fair coin 10 times?
Solution:Using binomial probability: P(X = 7) = C(10,7) x 0.5^7 x 0.5^3 C(10,7) = 10! / (7! x 3!) = 120 P = 120 x 0.0078125 x 0.125 = 0.1172 Probability = 11.72% Odds: about 1 in 8.5
Result:Exactly 7 heads: 11.72% probability | At least 7 heads: 17.19%
Example 2: Biased Coin - 6 Heads in 8 Flips
Problem:A coin is biased 60% toward heads. What is the probability of getting exactly 6 heads in 8 flips?
Solution:P(X = 6) = C(8,6) x 0.6^6 x 0.4^2 C(8,6) = 28 0.6^6 = 0.046656 0.4^2 = 0.16 P = 28 x 0.046656 x 0.16 = 0.2090 Probability = 20.90%
Result:Exactly 6 heads (biased coin): 20.90% | Expected heads: 4.8
Frequently Asked Questions
How do you calculate the probability of a specific number of heads in multiple coin tosses?
The probability of getting exactly k heads in n coin tosses follows the binomial probability distribution. The formula is P(X = k) = C(n,k) times p raised to the power k times (1-p) raised to the power (n-k), where C(n,k) is the binomial coefficient representing the number of ways to choose k successes from n trials, p is the probability of heads on a single toss (0.5 for a fair coin), and (1-p) is the probability of tails. For example, the probability of getting exactly 3 heads in 5 flips of a fair coin is C(5,3) times 0.5 to the fifth power, which equals 10 times 0.03125, giving a probability of 31.25 percent.
What is the difference between a fair coin and a biased coin?
A fair coin has an equal probability of landing on heads or tails, with each outcome having a 50 percent chance. In reality, perfectly fair coins do not exist due to slight asymmetries in weight distribution and design. A biased coin has unequal probabilities for heads and tails. For instance, a coin biased 60 percent toward heads will land on heads six out of ten times on average. The bias can be intentional, as in loaded coins used for demonstration purposes, or natural, as studies have shown that real coins may have subtle biases of around 51 percent toward the side facing up when flipped. Coin Toss Probability Calculator allows you to adjust the bias parameter to model both fair and biased coins accurately.
What is the expected value in coin tossing and how is it used?
The expected value represents the average number of heads you would get if you repeated the experiment many times. For n tosses of a coin with probability p of heads, the expected value is simply n times p. For 10 flips of a fair coin, the expected value is 5 heads. The standard deviation, calculated as the square root of n times p times (1-p), tells you how much variation to expect around the average. For 10 fair coin flips, the standard deviation is approximately 1.58, meaning most outcomes will fall between about 3 and 7 heads. These statistical measures help you determine whether observed results are consistent with a fair coin or suggest bias, which is the foundation of many hypothesis tests in statistics.
What is the gambler's fallacy and how does it relate to coin tossing?
The gambler's fallacy is the mistaken belief that past outcomes influence future independent events. If you flip a fair coin and get five heads in a row, the probability of heads on the next flip remains exactly 50 percent, not less. Each coin toss is an independent event with no memory of previous results. The confusion arises because over many flips, the proportion of heads tends to approach 50 percent (by the law of large numbers), which people misinterpret as meaning the coin must correct itself in the short term. In reality, the coin does not know or care what happened before. This fallacy has led to significant financial losses in gambling and poor decision-making in various real-world scenarios where people incorrectly assume random processes are self-correcting.
How many coin flips are needed to determine if a coin is fair?
Determining coin fairness requires statistical hypothesis testing, and the number of flips needed depends on the level of confidence desired and the magnitude of any bias present. As a general guideline, at least 100 flips are needed to detect a moderate bias with reasonable confidence. For detecting a small bias of around 55 percent, you would typically need 500 to 1,000 flips. The standard approach uses a binomial test or chi-squared test comparing observed results to the expected 50-50 split. If the observed proportion falls outside two standard deviations from 0.5, there is roughly 95 percent confidence that the coin is biased. For forensic or scientific applications, researchers often use 10,000 or more flips to achieve very high precision in estimating the true probability.
How do you calculate the probability of at least a certain number of heads?
An at-least probability is a cumulative binomial: sum the exact probabilities from your threshold up to n. For at least 8 heads in 10 fair flips, add the cases for 8, 9 and 10, whose combination counts are 45, 10 and 1, giving 56 favourable sequences out of the 1,024 possible sequences, about 5.47 percent. When the threshold is low it is faster to compute the complement and subtract from 1. Cumulative probabilities, not exact-count probabilities, are what hypothesis tests and p-values are built from.
How likely is a long run of heads in a sequence of flips?
A specific run of k heads starting at a fixed position has probability p^k, so five heads in a row on a fair coin is 0.5^5 = 1/32, about 3.13 percent. The chance of seeing such a run somewhere in a longer sequence is much higher, because every starting position offers another opportunity. This is why streaks that feel impossible show up routinely in a few hundred flips, and why judging fairness by eye from a short sequence is unreliable.
Does the law of large numbers mean heads and tails even out in absolute terms?
No. The law of large numbers says the proportion of heads converges toward p, not that the counts converge. The expected absolute gap between heads and tails actually grows with the number of flips, roughly in proportion to the square root of n, while that same gap divided by n shrinks toward zero. So a coin can be 40 flips ahead on heads after ten thousand tosses and still show a proportion very close to 0.5. Confusing the two statements is the mathematical root of the gambler's fallacy.
When can you use the normal approximation instead of the binomial formula?
For large n the binomial distribution is closely approximated by a normal curve with mean n times p and standard deviation the square root of n times p times (1-p). A common working rule is that both n times p and n times (1-p) should be at least 10. With 100 fair flips the mean is 50 and the standard deviation is 5, so about 95 percent of results fall between 40 and 60 heads. Apply a continuity correction of half a unit when converting a discrete count into a normal interval, and fall back to the exact binomial for small n or extreme p.
Can you get a fair result from a biased coin?
Yes, using von Neumann's trick. Flip the coin twice and ignore any pair that comes out HH or TT. If the pair is HT, call it heads; if it is TH, call it tails. Because the two flips are independent, HT and TH both have probability p times (1-p) and are therefore exactly equally likely no matter what the bias is. The cost is efficiency: pairs are discarded whenever both flips match, so the more biased the coin, the more flips each fair bit consumes.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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