Question

 Four coins are tossed. Let Z be the random variable representing the number heads that occur. Find the values of the random variable Z. ( Take note that you should have 2 = 16 possible outcomes ).

Answer 1

There are $2^4=16$ possible outcomes

$$S=\{HHHH, HHHT, HHTH, HTHH, THHH,$$

$$HHTT, HTHT, HTTH, THTH, THHT, TTHH,$$

$$HTTT, THTT, TTHT, TTTH, TTTT\}$$

The possible values of the random variable $Z$ are $0, 1, 2, 3, 4.$

We will assume that the probability of getting heads and tails is the same:

$$p = q =1/2$$ $$\def\arraystretch{1.5} \begin{array}{c:c} Possible \ Outcomes & Z \\ \hline HHHH & 4 \\ \hdashline HHHT & 3 \\ \hdashline HHTH & 3 \\ \hdashline HTHH & 3 \\ \hdashline THHH & 3 \\ \hdashline HHTT & 2 \\ \hdashline HTHT & 2 \\ \hdashline HTTH & 2 \\ \hdashline THTH & 2 \\ \hdashline THHT & 2 \\ \hdashline TTHH & 2 \\ \hdashline HTTT & 1 \\ \hdashline THTT & 1 \\ \hdashline TTHT & 1 \\ \hdashline TTTH & 1 \\ \hdashline TTTT & 0 \\ \hdashline \end{array}$$

Construct the probability distribution of the random variable

$$\def\arraystretch{1.5} \begin{array}{c:c} z & 0 & 1 & 2 & 3 & 4 \\ \hline p(z) & 1/16 & 1/4 & 3/8 & 1/4 & 1/16 \end{array}$$

or

$$\def\arraystretch{1.5} \begin{array}{c:c} z & 0 & 1 & 2 & 3 & 4 \\ \hline p(z) & 0.0625 & 0.25 & 0.375 & 0.25 & 0.0625 \end{array}$$