Coin toss probability is an excellent introduction to the basic principles of probability theory because a coin has a mostly equal chance of landing heads or tail. So, a coin toss is a popular and fair method of making an unbiased decision. Here is a look at how coin toss probability works, with the formula and examples.

- When you toss a coin, the probability of getting heads or tails is the same.
- In each case, the probability is ½ or 0.5. In other words, “heads” is one of two possible outcomes. The same is true for tails.
- Find probability of multiple independent events by multiplying the probability of individual events. For example, the probability of getting heads and then tails (HT) is ½ x ½ = ¼.

### The Basics of Coin Toss Probability

A coin has two sides, so there are two possible outcomes of a fair coin toss: heads (H) or tails (T).

#### Coin Toss Probability Formula

The formula for coin toss probability is the number of desired outcomes divided by the total number of possible outcomes. For a coin, this is easy because there are only two outcomes. Getting heads is one outcome. Getting tails is the other outcome.

**P = (number of desired outcomes) / (number of possible outcomes)**

P = 1/2 for either heads or tails

The probability of getting either heads or tails (2 possible outcomes) is 1. In other words, when you toss a coin you are pretty much guaranteed to get either heads or tails.

P = 2/2 = 1

Getting heads or tails on a coin are **mutually exclusive events**. If you get heads, you don’t get tails (and vice versa). Another way of calculating the probability of two mutually exclusive events is adding their individual probabilities. For one coin toss:

P(heads or tails) = ½ + ½ = 1

#### Probability for Multiple Coin Tosses

If you toss a coin more than once and want the probability of a specific outcome, you multiply the probability values of each toss. This works when the tosses are **independent events**. What this means is the outcome of the second toss (or third, etc.) is not dependent on the outcome of the first toss (or any other previous or subsequent toss).

For example, let’s calculate the probability of getting heads, heads, tails (HHT):

P(HHT) = ½ x ½ x ½ = ⅛

### Coin Toss Probability Example Problems

Coin toss problems usually are word problems. The key is understanding what the problem is asking.

For example, calculate the probability of tossing a coin twice times and getting at least one “heads”.

#### Solution

First, write down all the possible outcomes of randomly tossing a coin three times:

HH, HT, TH, TT

There are four possible outcomes.

Next, determine how many of these outcomes are “favorable outcomes” or ones that meet the criteria in the problem. There are three outcomes where at least one toss has a “heads” result.

Now, perform the calculation:

P = favorable outcomes / total outcomes

P (at least one H) = 3/4 or 0.75

Now, what is the probability of both tosses showing the same face? In other words, what is the chance of both tosses showing heads or both showing tails?

#### Solution

Again, you have four possible outcomes. There are two favorable outcomes (HH or TT).

P (both heads or both tails) = 2/4 = 1/2 or 0.5

### What Is a Fair Coin?

A “fair coin” is one which has an equal probability of landing heads or tails in a coin toss. In contrast, an unfair coin is one which is weighted or filed so that it has a greater chance of landing on one side than the other.

In practice, most coins are not totally fair because the raised metal slightly favors one side (on the order of 0.49 to 0.51). Also, for an ordinary person, there is a slight bias that favors catching a coin in the same orientation as how it was thrown (0.51). Skilled conjurers and gamblers can toss or catch a coin so that it lands with considerable bias, even if the coin is fair.

There is also a slight chance of a coin landing on its edge. For example, an American nickel lands on its edge about 1 in 6000 tosses.

### Randomness and Probability

Even though a fair coin has even odds of a heads or tails result, the outcome is random. So, if you toss a coin twice, probability calculates you only have a 1 in 4 chance of getting HH. If you repeat the process and toss the coin two more times, you can get different results. The *probable* outcome becomes more likely the more times you repeat the process.

With this in mind, do you think a coin is biased if it is tossed a certain number of times and 3/4 (75%) of the time it was heads? The answer is that you cannot make a determination of fairness, because you don’t know whether the coin was tossed four times or four thousand times! If, however, you know the number of tosses, you have a real sense of whether or not a coin is fair.

### References

- Ford, Joseph (1983). “How random is a coin toss?”.
*Physics Today*. 36 (4): 40–47. doi:10.1063/1.2915570 - Kallenberg, O. (2002)
*Foundations of Modern Probability*(2nd ed.). Springer Series in Statistics. ISBN 0-387-95313-2. - Murray, Daniel B.; Teare, Scott W. (1993). “Probability of a tossed coin landing on edge”.
*Physical Review E*. 48 (4): 2547–2552. doi:10.1103/PhysRevE.48.2547 - Vulovic, Vladimir Z.; Prange, Richard E. (1986). “Randomness of a true coin toss”.
*Physical Review A*. 33 (1): 576–582. doi:10.1103/PhysRevA.33.576