A statistics quiz consists of 10 multiple questions. There are four choices for each question.
One student is unprepared and decides to guess the answers to every question. Assuming 70%
is a passing grade, find the probability that the student will pass the quiz.
Let's work on this together.
1. How many questions does he need to guess right to pass the test? (include all possible values)
2. What kind of distribution does this problem fall into?
gas
To find the probability that the student will pass the quiz, we need to determine the minimum number of correct answers required for a passing grade.
Since there are 10 questions and four choices for each question, the probability of correctly guessing any single question is 1/4.
To calculate the probability of getting a specific number of correct answers in a multiple-choice quiz, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k correct answers,
- (n C k) represents the number of ways to choose k correct answers out of n questions,
- p is the probability of getting a single question correct, and
- n is the total number of questions.
In this case, k represents the number of correct answers needed for a passing grade. Since 70% is considered a passing grade, the student needs to score at least 7 out of 10 correct.
Now we can calculate the probability of the student passing by summing the probabilities of getting 7, 8, 9, or 10 questions correct:
P(pass) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Let's calculate the probabilities for each case:
P(X = 7) = (10 C 7) * (1/4)^7 * (3/4)^(10-7)
P(X = 8) = (10 C 8) * (1/4)^8 * (3/4)^(10-8)
P(X = 9) = (10 C 9) * (1/4)^9 * (3/4)^(10-9)
P(X = 10) = (10 C 10) * (1/4)^10 * (3/4)^(10-10)
Now we can calculate these probabilities and add them to get the final answer.