Question

Five 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

We will assume that the probability of getting heads and tails is the same: $p = q = \frac{1}{2}$.

Using Bernoulli's formula, we find the probability that 0, 1, 2, 3, 4, and 5 heads will land:

$P(0) = {q^5} = {\left( {\frac{1}{2}} \right)^5} = \frac{1}{{32}}$

$P(1) = C_5^1p{q^4} = 5 \cdot {\left( {\frac{1}{2}} \right)^5} = \frac{5}{{32}}$

$P(2) = C_5^2{p^2}{q^3} = 10 \cdot {\left( {\frac{1}{2}} \right)^5} = \frac{{10}}{{32}}$

$P(3) = C_5^3{p^3}{q^2} = 10 \cdot {\left( {\frac{1}{2}} \right)^5} = \frac{{10}}{{32}}$

$P(4) = C_5^4{p^4}q = 5 \cdot {\left( {\frac{1}{2}} \right)^5} = \frac{5}{{32}}$

$P(5) = {p^5} = {\left( {\frac{1}{2}} \right)^5} = \frac{1}{{32}}$

We get the distribution law

$\begin{matrix} Z&0&1&2&3&4&5\\ p&{\frac{1}{{32}}}&{\frac{5}{{32}}}&{\frac{{10}}{{32}}}&{\frac{{10}}{{32}}}&{\frac{5}{{32}}}&{\frac{1}{{32}}} \end{matrix}$