Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W.

Solution:

The sample space S for the three tosses of the coin is:

$S=\{{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} \}$

Let $W$ be a random variable giving the number of heads minus the number of tails in three tosses of a coin, we assign a value of $\omega$ of $W$ to each sample point in the following way: 

$\begin{array}{cc} \text{Sample points} & & \omega \\ HHH & & 3 \\HHT & & 1 \\HTH & & 1 \\HTT & & -1 \\THH & & 1 \\THT & & -1 \\TTH & & -1\\ TTT & & -3 \end{array}$

The sample space for $W$ is

$S_W=\{{-3, -1, 1, 3} \}$

corresponding to $3T,1H2T,2H1T,$ and $3H$ respectively.

$P(0\ heads\ \&\ 3\ tails)=\binom{3}{0}({1 \over 2})^0({1 \over 2})^3={1 \over 8}$$P(1\ head\ \&\ 2\ tails)=\binom{3}{1}({1 \over 2})^1({1 \over 2})^2={3 \over 8}$$P(2\ heads\ \&\ 1\ tail)=\binom{3}{2}({1 \over 2})^2({1 \over 2})^1={3 \over 8}$$P(3\ heads\ \&\ 0\ tails)=\binom{3}{3}({1 \over 2})^3({1 \over 2})^0={1 \over 8}$$P(W=-3)={1 \over 8}$$P(W=-1)={3 \over 8}$$P(W=1)={3\over 8}$

$P(W=3)={1 \over 8}$