91Ó°ÊÓ

Probability is the state of being able to do something. It's a branch of mathematics that deals with the randomness of events. From zero to one, the value is stated. In mathematics, the concept of probability was established to predict the likelihood of occurrences occurring. Probability simply refers to the likelihood of something occurring.

The described random variables have a binomial distribution. Let X be the random binomial variable

\({n_1} = 200{\rm{ and }}{p_1} = 0.45\) (for the rural voters), and let Y be a binomial random variable with parameters \({n_2} = 300\;and\;{p_2} = 0.6\)(for suburban and urban).

Since approximate probability is needed, notice that \({n_1}\;and\;{n_2}\)are large samples, which indicates that X and Y are approximately normal.

Proposition: For a binomial random variable X with parameters \(n,\;p,\;and\;q = 1 - p\), the following is true

\(\begin{array}{l}E(X) = np\\V(X) = np(1 - p) = npq\\{\sigma _X} = \sqrt {npq} \end{array}\)

The mean value of approximate normal random variable X is

\({n_1} \cdot {p_1} = 200 \cdot 0.45 = 90\)

and he mean value of approximate normal random variable Y is

\({n_2} \cdot {p_2} = 300 \cdot 0.6 = 180\)

The random variable of interest is X+Y, the total number of voters who vote for a particular gubernatorial candidate. The expected value of the random variable is

\(E(X + Y) = E(X) + E(Y) = 90 + 180 = 270,\)

and the variance is

\(\begin{aligned} V(X + Y) &= V(X) + V(Y)\\ &= {n_1} \cdot {p_1} \cdot \left( {1 - {p_1}} \right) + {n_2} \cdot {p_2} \cdot \left( {1 - {p_2}} \right)\\ &= 200 \cdot 0.45 \cdot 0.55 + 300 \cdot 0.6 \cdot 0.4\\ &= 121.4\end{aligned}\)

Here we used the independence of random variables and the variance of Binomial random variable

The standard deviation of random variable X+Y is

\({\sigma _{X + Y}} = \sqrt {121.4} = 11.02\)

Now by standardizing random variable X+Y the approximate probability can be calculated. The following is true (with the continuity correction)

\(\begin{aligned}P(X + Y \ge 250) &= P\left( {\frac{{X + Y - E(X + Y)}}{{{\sigma _{X + Y}}}} \ge \frac{{249.5 - 270}}{{11.02}}} \right)\\ &= P(Z \ge - 1.86)\\ &= 1 - P(Z < - 1.86)\mathop = \limits^{(1)} 1 - 0.0424\\ &= 0.9686\end{aligned}\)

(1): from the normal probability table in the appendix. The probability can also be computed with a software.