A class is given a list of eight study problems from which five will be part of an upcoming exam. If a given student knows how to solve six of these problems, find the probability that the student will be able to answer:
(a) all five questions on the exam.
(b) exactly eight questions on the exam.
(c) at least nine questions on the exam.
No matter what I try -- permutations, combinations, random multiplying and dividing -- I can't figure any of these parts out. I don't even know where to start, really.
To calculate the probability of answering a certain number of questions on the exam, we need to use the concept of combinations.
In this case, we have a total of 8 study problems, out of which the student knows how to solve 6. Since only 5 questions will be part of the exam, we can approach this problem using combinations.
(a) To calculate the probability that the student will be able to answer all five questions correctly, we need to consider the combination of selecting 5 questions out of the 6 known problems that the student can solve.
The probability can be calculated using the formula:
P(all five questions) = (Number of ways to select 5 questions the student knows) / (Number of ways to select any 5 questions from 8)
Number of ways to select 5 questions the student knows = C(6, 5) = 6
Number of ways to select any 5 questions from 8 = C(8, 5) = 56
So, P(all five questions) = 6/56 = 3/28
(b) To calculate the probability that the student will be able to answer exactly eight questions, we need to consider the combination of selecting all 6 questions the student knows and then selecting 2 additional questions from the remaining 2 questions that the student doesn't know.
The probability can be calculated using the formula:
P(exactly eight questions) = (Number of ways to select 6 known questions) * (Number of ways to select 2 unknown questions) / (Number of ways to select 5 questions from 8)
Number of ways to select 6 known questions = C(6, 6) = 1
Number of ways to select 2 unknown questions = C(2, 2) = 1
Number of ways to select 5 questions from 8 = C(8, 5) = 56
So, P(exactly eight questions) = (1 * 1) / 56 = 1/56
(c) To calculate the probability that the student will be able to answer at least nine questions, we need to consider the combination of selecting all 6 questions the student knows and selecting all 3 remaining questions from the remaining 2 questions that the student doesn't know.
The probability can be calculated using the formula:
P(at least nine questions) = (Number of ways to select 6 known questions) * (Number of ways to select 3 unknown questions) / (Number of ways to select 5 questions from 8)
Number of ways to select 6 known questions = C(6, 6) = 1
Number of ways to select 3 unknown questions = C(2, 3) = 0 (since there are only 2 unknown questions available)
Number of ways to select 5 questions from 8 = C(8, 5) = 56
So, P(at least nine questions) = (1 * 0) / 56 = 0
Therefore, the probability of answering all five questions is 3/28, the probability of answering exactly eight questions is 1/56, and the probability of answering at least nine questions is 0.