Question

Common names The Census Bureau says that the $10$ most common names 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. Out of curiosity, you look at the authors of the textbooks for your current courses. There are $9$ authors in all. Would you be surprised if none of the names of these authors were among the $10$most common? (Assume that authors’ names are independent and follow the same probability distribution as the names of all residents.)

Short answer

$P(9notcommon)=40.32\%$ not surprising

Step-by-step solution

Step 1. Given Information

The top ten most frequent names in the United States account for$9.6\%$ of all residents.

Step 2. Concept used

Multiplication rule: $P(A\cap B)=P(A\cap B)\times P(B/A)$

Complement rule: $P(notA)=1-P\left(A\right)$

Step 3. Calculation

$P\left(common\right)=9.6\%=0.096$

Complement rule:

$P(notA)=1-P\left(fail\right)=1-0.096=0.904$

Multiplication rule:

$P(A\cap B)=P\left(A\right)\times P\left(B\right)$

Then it can calculate the likelihood of 9 uncommon names:

$P(9notcommon)=(P(notcommon\left)\right)9=(0.94)9\approx 0.4032=40.32\%$

Because the probability is larger than $5\%$, there is a good chance of getting 9 uncommon names, which is not unusual.