Question

What is the expected standard deviation of a single coin flip, where heads = 1 and tails = 0 ?

Answer 1

Hint: We must evaluate the given experiment and decide which type of distribution is given here. We also know that in a binomial distribution, the mean = np and variance = npq. We can find the standard deviation by taking the square root of variance.

Complete step-by-step answer:
Here, the event is tossing a coin. So, there can be only two possible outcomes, which are heads and tails.
We know that when there are only two outcomes possible, then the distribution is called binomial distribution. We can also define a binomial distribution as the probability of a success or failure outcome in an experiment or survey that has been repeated a number of times.
If we consider the probability of success to be $p$ and the probability of failure to be $q$, then we can clearly see that $q=1-p$.
If the event is repeated n number of times, or n observations are taken, then we know that the mean, variance and standard deviation of this experiment can be calculated as
Mean = $np$
Variance = $npq$
Standard deviation = $\sqrt{npq}$
Here, in this problem, a coin is flipped once. So, we have n = 1.
In the question, we are given that heads = 1 and tails = 0. Thus, we can consider heads as a success and tails as a failure.
So, we can say that the probability of a success is the probability of getting a head. Since, the probability of getting a head is one half, we can say that
$p=\dfrac{1}{2}$
And thus, $q=\dfrac{1}{2}$.
Hence, variance = $1\times \dfrac{1}{2}\times \dfrac{1}{2}$
Therefore, variance = $\dfrac{1}{4}$.
So, standard deviation = $\sqrt{\text{Variance}}=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}$.
Thus, the standard deviation of this experiment is $\dfrac{1}{2}$.

Note: We must take care not to consider the information given in the question that heads = 1 and tails = 0 as the probability of heads = 1 and the probability of tails = 0. Heads = 1 simply means that getting a head is success and tails = 0 means that getting a tail is failure.