Question

four coins are tossed. let z be the random variable representing the number of heads that occur. find the values of the random variable z.

Answer 1

Four coins are tossed. For each toss, the outcome belongs to one of the following:

$$\{Head, Tail\}$$

We want to count the total number of heads obtained from those 4 tosses. $Z$ is the random variable representing the number of heads that occur. We want to find the all possible values of $Z$.

Observe that, any one of the following cases may happen:

1. All the four outcomes are 'Tail'. So number of heads is 0. Hence, $Z=0$ in this case.
2. Any 3 of the 4 outcomes are 'Tail' and the remaining one is a 'Head' Hence, $Z=1$ in this case.
3. Any 2 of the 4 outcomes are 'Tail' and the remaining two are 'Head' Hence, $Z=2$ in this case.
4. Any 1 of the 4 outcomes is 'Tail' and the remaining three are 'Head' Hence, $Z=3$ in this case.
5. All the four outcomes are 'Head'. So number of heads is 4. Hence, $Z=4$ in this case.

Note that, the 5 cases listed above explores all possible outcomes. Hence, the random variable $Z$ takes any one value from $\{0,1,2,3,4\}$.