How to generate a random number between 0 and 7 using a coin

Generate a random number

While generating a random number from a given range, we must ensure that all numbers have an equal likelihood of being generated. For example, when generating a random number between $0$ and $7$, each of the numbers $0, 1, 2, 3, 4, 5, 6,$ and $7$ has a $\frac{1}{8}$ or $0.125$ chance of being generated.

We can think of a coin as a random number generator. Flipping a coin only has two outcomes, heads or tails, each with a probability of $0.5$.

To use a coin to generate a number between $0$ and $7$, we must first understand the binary representation of numbers.

Binary number system

Numbers in our everyday life are usually in the decimal or base-10 system because each digit can have one of ten values ($0–9$). In the binary or base-2 system, each digit can only have one of two values, $0$ or $1$.

We can use at least three digits to represent all the numbers in the range $[0,7]$.

Note: Click here to learn how to convert decimal and binary values.

As we can see, a base-10 number can be represented as a sequence of 1s and 0s. If we have to generate a base-10 number between $0$ and $7$, we can generate three binary numbers instead.

Luckily, we have a random binary generator, our coin!

Example

If we flip the coin three times and note the results as $0$ or $1$, we get a three-digit binary number. Once we convert this binary number to base-10, we'll get a random number between $0$ and $7$.

The simulation below demonstrates the process:

Equal probabilities

As we stated before, any number between $0–7$ should appear with the probability of $0.125$.

Three consecutive coin flips have the following possible outcomes. An 'H' represents heads while a 'T' represents tails.

Note: We treat a heads as 0 and a tails as 1.

Any single outcome has a 1 in 8 chance of occurring. Hence each number is equally likely.

We can use a coin can to generate a random number between any range $[0,\ 2^n-1]$where $n\geq1$. The number of coin-flips and digits of the resulting binary number would be $n$. With $n=3$ we generated a number in $[0, 7]$ with $3$ coin-flips and a $3$ digit resultant binary number.