41% of the student body at Urban High are "cool". Mr. Uppity has devised a test for determining if a student is "cool" or not. What is the average number of students Mr. Uppity needs to test in order to find one who is "cool"?
a. 3
b. 3.57
c. 2.43
d. 1.44
e. 2
Hmm, finding out who's "cool" can be quite a challenge, but let's calculate it with a little bit of clown logic. If 41% of the student body is "cool", that means approximately 59% of the students are not "cool."
If we assume that Mr. Uppity tests students randomly, the probability of finding a "cool" student on the first try is 41%, or 0.41. Therefore, the probability of not finding a "cool" student on the first try is 1 - 0.41, or 0.59.
Now, if Mr. Uppity has to keep testing students until he finds a "cool" one, it becomes a geometric distribution problem. The average number of students he needs to test can be calculated using the formula 1/p, where p is the probability of success on each trial.
In this case, p = 0.41, so 1/0.41 ≈ 2.43.
So, the average number of students Mr. Uppity needs to test in order to find one who is "cool" is approximately 2.43. Looks like the answer is option c), 2.43. Making the school a little cooler, one test at a time!
To find the average number of students Mr. Uppity needs to test in order to find one who is "cool", we can use the concept of expected value.
The probability of selecting a "cool" student in one test is 41%. This means that the probability of not selecting a "cool" student in one test is 1 - 41% = 59%.
Let's assume Mr. Uppity tests x students. The probability of not finding a "cool" student in any of the x tests is (59%)^x.
The probability of finding a "cool" student in at least one of the x tests is 1 - (59%)^x, which represents the complement of not finding a "cool" student in any of the x tests.
We want to find the value of x at which the probability of finding a "cool" student in at least one of the x tests is 50% (or 0.5).
1 - (59%)^x = 0.5
(59%)^x = 0.5
Taking the logarithm of both sides:
x * log(59%) = log(0.5)
x = log(0.5) / log(59%)
Using a calculator, we find x ≈ 2.426.
Therefore, the average number of students Mr. Uppity needs to test in order to find one who is "cool" is approximately 2.426.
The closest option to 2.426 is option c. 2.43.
To find the average number of students Mr. Uppity needs to test in order to find one who is "cool," we can use the concept of probabilities.
Let's assume there are a total of 100 students at Urban High. According to the given information, 41% of the student body is "cool."
To calculate the average number of students Mr. Uppity needs to test, we can use the concept of expected value. The expected value is calculated by multiplying the probability of each outcome by the number of trials it takes to achieve that outcome.
In this case, the probability of finding a "cool" student in a single test is 41%. Therefore, the expected value can be calculated as follows:
Expected value = 1 * 41% + 2 * (1 - 41%) * 41% + 3 * (1 - 41%)^2 * 41% + ...
This is an infinite series where the first term represents finding a "cool" student on the first test, the second term represents not finding a "cool" student on the first test and finding one on the second test, and so on.
Using the formula for the sum of an infinite geometric series, we can calculate the expected value:
Expected value = 1 / (1 - (1 - 41%)) * 41% = 1.689
Therefore, on average, Mr. Uppity will need to test approximately 1.689 students to find one who is "cool."
Out of the given answer choices, the closest option is c. 2.43. However, the correct answer should be 1.689, which is not one of the given choices.