Question

Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randomly selected​ (with replacement). Find the probability that for one such phone​ number, the last four digits include at least one 0.

Answer 1

The correct answer is:

0.3439.

Explanation:

This is a binomial probability. This is because there are two outcomes; either a digit is 0 or it is not. Since the digits are used with replacement, the probability of one digit being a 0 does not affect the probability of a second digit being 0. Finally, there are a fixed number of trials; we are choosing 4 digits.

Since we want the probability that at least one digit is a zero, we want the complement of "no digits are zero". This means we will find P(X = 0) and subtract it from 1.

The formula for a binomial distribution with n trials and r successes is

$P(X=r)=_nC_r(p)^r(1-p)^{n-r}$

In this situation, n is 4, since we are choosing 4 digits. r is 0, since we are finding the probability that no digits are 0. p is 0.1, since there is a 1/10 chance of a digit being 0. This gives us:

$P(X=0)=_4C_0(0.1)^0(1-0.1)^l{4-0} \\ \\=\frac{4!}{0!4!}(0.1)^0(0.9)^4 \\ \\=1(1)(0.9^4)=0.6561$

This means that the complement of P(X=0), or the probability that at least one digit is 0, is 1-0.6561 = 0.3439.

Answer 2

Values are 0to1
n=10
1-(4/10)^4=0.344