Question

4.112 California’s electoral college votes. During a presidential election, each state is allotted a different number of votes in the Electoral College, depending on the population. For example, California is allotted 55 votes (the most) while several states (including the District of Columbia) are allotted 3 votes each (the least). When a presidential candidate wins the popular vote in a state, the candidate wins all the Electoral College votes in that state. To become president, a candidate must win 270 of the total of 538 votes in the Electoral College. Chance(Winter 2010) demonstrated the impact on the presidential election of winning California. Assuming a candidate wins California’s 55 votes, the number of additional Electoral College votes the candidate will win can be approximated by a normal distribution with $\mathit{\mu }\mathbf{=}\mathbf{241}\mathbf{.}\mathbf{5}$votes and $\mathit{\sigma }\mathbf{=}\mathbf{49}\mathbf{.}\mathbf{8}$votes. If a presidential candidate wins the popular vote in California, what are the chances that he or she becomes the next U.S. president?

Short answer

The probability of he or she becoming the next U.S president is 0.2836.

Step-by-step solution

Given information

Assuming a candidate wins California’s 55 votes, the number of additional Electoral College votes the candidate will win can be approximated by a normal distribution with mean 241.5 votes and standard deviation 49.8 votes.

Calculating the probability

Let x be the random variable denoting Electoral College votes.

Here x is normally distributed with mean 241.5 votes and standard deviation 49.8 votes.

$x~N(241.5,49.8)$

If a presidential candidate wins the popular vote in California then the probability of he or she becoming the next U.S president has been calculated below.

To become president a candidate must have to win 270 votes.

$$\begin{gathered} P(x\ge 270)=1-P(x<270) \\ =1-P\left(\frac{x-241.5}{49.8}<\frac{270-241.5}{49.8}\right) \\ =1-P(z<0.572289) \\ =1-0.716437 \\ =0.283563 \\ =0.2836 \\ P(x\ge 270)=0.2836 \end{gathered}$$

So, the probability of he or she becoming the next U.S president is 0.2836.