Question

What is the expected value of a fair coin toss, 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. We can also find the mean or expected value by taking the sum of all possible outcomes, that is 0 and 1, and then dividing by the number of possibilities, that is, 2.

Complete step-by-step answer:
Here, we know that the event is tossing a fair coin. So, there can be only two possible outcomes, heads or tails.
We are very well aware 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 any number of times.
Let us consider the probability of success to be $p$ and the probability of failure to be $q$. Then we can clearly say that $q=1-p$.
We know that if the event is repeated n number of times, or n observations are taken, then the mean, variance and standard deviation of this experiment can be calculated as
Mean or Expected value = $np$
Variance = $npq$
Standard deviation = $\sqrt{npq}$
Here, in this problem, a fair coin is flipped once. So, we have n = 1.
Also in the question, we are given that heads = 1 and tails = 0. Thus, we need to consider heads as a success and tails as a failure.
So, we can say that the probability of success is same as the probability of getting a head. Since, we know very clearly that the probability of getting a head is one half, we can say that $p=\dfrac{1}{2}$.
And thus, $q=\dfrac{1}{2}$.
Hence, mean or expected value = $1\times \dfrac{1}{2}$
Therefore, Expected value = $\dfrac{1}{2}$.
Thus, the expected value 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. Here, heads = 1 simply means that getting a head is success and tails = 0 means that getting a tail is failure.