Question

Four coins are tossed. Let Y be the random variable that represents the number tail that will occur. Find the values of random variable Y.

Answer 1

The possible number of tails is: 0,1,2,3,4. Those are the values that $Y$ can take.

In order to compute the respective probabilities we will use the binomial distribution. We assume that the probability of a tail and the probability of a head are equal. We receive:

$P(Y=0)=\left(\frac{1}{2}\right)^4=\frac{1}{16};P(Y=1)=C_4^1\left(\frac{1}{2}\right)^4=4\left(\frac{1}{2}\right)^4=\frac{1}{4};$

$P(Y=2)=C_4^2\left(\frac{1}{2}\right)^4=\frac{3\cdot4}{2}\left(\frac{1}{2}\right)^4=\frac{3}{8}; P(Y=3)=C_4^3\left(\frac{1}{2}\right)^4=\frac{1}{4}$;

$P(Y=4)=C_4^4\left(\frac{1}{2}\right)^4=\frac{1}{16}$

We check that the sum of all probabilities is 1: $P(Y=0)+P(Y=1)+P(Y=2)+P(Y=3)+P(Y=4)=$

$\frac{1}{16}+\frac{1}{16}+\frac{1}{4}+\frac{1}{4}+\frac{3}{8}=\frac{1}{2}+\frac{1}{2}=1$