Question

In a population where 77% of voters prefer Candidate A, an organization conducts a poll of 19 voters. Find the probability that 14 of the 19 voters will prefer Candidate A.
(Report answer accurate to 4 decimal places. That is, round to 4 decimal places.)

Answer 1

Answer:

$P(X=14)=(19C14)(0.77)^{14} (1-0.77)^{19-14}=0.1928$

Then the probability that 14 of the 19 voters will prefer Candidate A is approximately 0.1928 or 19.28%

Step-by-step explanation:

We can define X the random variable of interest "number of voters that will prefer Candidate A", since we have a sample size given and a probability of success we can use the binomial distribution to model the random variable. And on this case we can assume the following distribution:

$X \sim Binom(n=19, p=0.77)$

The probability mass function for the Binomial distribution is given by:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

For this problem we want to find this probability:

$P(X=14)$

And usign the probability mass function defined before we got:

$P(X=14)=(19C14)(0.77)^{14} (1-0.77)^{19-14}=0.1928$

Then the probability that 14 of the 19 voters will prefer Candidate A is approximately 0.1928 or 19.28%