Answer to Question #169390 in Statistics and Probability for John

Question

1.     Suppose that U is a random variable giving the number of tails minus the number of heads in three (3) tosses of a coin. List the elements of the sample space S for three (3) tosses of the coin and to each sample point assign a value u of U. 

Answer 1

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

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

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

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

The sample space for $U$ is

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

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

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

$P(2heads \&\ 1 tails)= \binom{3}{1}({1\over 2})^1({1\over2})^2={3\over8}$

$P(1\ heads \& \ 2\ t ails)=\binom{3}{2}({1\over 2})^2({1\over 2})^1={3\over 8}$

$P(0\ heads \& \ 3\ t ails)=\binom{3}{3}({1\over 2})^3({1\over 2})^0={1\over 8}$

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