Question

The Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Jones, Brown, Jones, Miller, Davis, Garcia, Rodriguez, Wilson. These names account for 9.6% of all U.S. residents.

Out of curiosity, you look at the authors of the textbooks for your current courses. There are 11 authors in all. Would you be surprised if none of the names of these authors were among the 10 most common?
 Give a probability to support your answer. Your answer should be rounded to three decimal places

Answer 1

The probability that all 11 textbook authors have names not in the 10 most common US names is approximately 0.382, so it's not especially surprising if none have a common name.

Step-by-step explanation:

The probability that none of the 11 authors have a name in the list of the 10 most common US names can be calculated using the complement rule. Given that 9.6% of US residents have one of those common names, each author independently has a 0.096 chance of having a common name, and a 1 - 0.096 = 0.904 chance of not having a common name.

To find the probability that all 11 authors have unique names (i.e., not in the 10 common names), we would multiply the individual probabilities for each author:

Probability = 0.90411

Calculating this gives us:

Probability ≈ 0.382

Since the probability is significantly higher than zero, it would not be particularly surprising if none of the authors had a common name.

Answer 2

Using binomial distribution where success is the appearing of any of the top 10 most common names, thus probability of success (p) is 9.6% = 0.096 and the probability of failure = 1 - 0.096 = 0.904. Number of trials is 11.
Binomial distribution probability is given by P(x) = nCx (p)^x (q)^(n - x)
Probability that none of the top 10 most common names appears is P(0) = 11C0 (0.096)^0 (0.904)^(11 - 0) = (0.904)^11 = 0.3295
Thus, the probability that at least one of the 10 most common names appear is 1 - 0.3295 = 0.6705

Therefore, I will be supprised that none of the names of the authors were among the 10 most common names given that the probability that at least one of the names appear is 67%.