Question

The Census Bureau says that the 10 most common surnames in the United States are, in order, Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez, and Wilson. These names account for 9.6% of all U.S. residents. Suppose you look at the authors of the textbooks for your current courses and find that there are 8 authors in total. What is the probability that none of the 8 surnames of these authors were among the 10 most common

Answer 1

0.4460 = 44.60% probability that none of the 8 surnames of these authors were among the 10 most common

Explanation:

For each surname, there are only two possible outcomes. Either they are among the 10 most common, or they are not. Each author is independent of each other. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

These names account for 9.6% of all U.S. residents.

This means that
`p = 0.096`

8 authors in total.

This means that
`n = 8`

What is the probability that none of the 8 surnames of these authors were among the 10 most common

This is
`P(X = 0)`. So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(8,0).(0.096)^(0).(0.904)^(8) = 0.4460`

0.4460 = 44.60% probability that none of the 8 surnames of these authors were among the 10 most common