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.

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

[tex] P(X=r)=_nC_r(p)^r(1-p)^{n-r} [/tex]

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:

[tex] 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 [/tex]

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.

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.

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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.