Question

Two coins are tossed at the same time. Let random variable be the number of heads showing.

a) Construct a probability distribution for

b) Find the expected value of the number of heads.

Answer 1

2 coins are tossed at the same time.

Let, X be the random variable, denoting the number of heads showing.

Now, we know that the probability of a head showing up, is 1/2, and the probability of head not showing up, is (1-1/2), ie. 1/2.

So, the probability of success is 1/2, and the probability of failure is 1/2.

The 2 tosses are independent also.

So, we can safely say that X follows binomial distribution with n=2, and p=1/2.

(a) So, the probability mass function of X is given by

Where, i can take integer values 0, 1, and 2.

So, the probability distribution of X is binomial with n=2 and p=1/2; and the probability mass function is given by

Where,

Now, we find the corresponding probabilities.

So, the probability distribution is

When   

   When

   When   

(b) Now, we have to find the expected number of heads; ie. we have to find the mean of X.

So, we have to find

.

Now, we know that

So, the expected number of heads is 1.

Note:- The mean value can also be identified straightaway, as the mean of binomial(n,p), is the product of n and p.

Here, n=2 and p=1/2. So, n*p=1.