a multiple choice quiz has 10 questions. each question has five possible answers.carman is certain that she knows the correct answers for questions 1 and 4. if she guesses the other questions, determine the probability that she gets 60 % on the quiz ?
i know this question involves binomial distribution but im not sure with the correct format and order that im supposed to put the question in ?
she needs at least 4 out of the remaining 8 questions correct. So, that has probability
This topic is discussed here:
http://math.stackexchange.com/questions/853407/4-heads-in-8-tosses
The probability of any one getting correct with five possible answers is .2, and to make a 60 she has to get four more right.
P(4 correct)=8C4*(.2)^4(.8)^4=
= 8*7*6*5/4*3*2*1 (.2)^4 * (.8)^4
= 7*2*5 * (.2)^4 (.8)^4
= 70 * 0.00065536
= 0.0458752
Now if you wanted to calculate the probability of getting at least a 60 , then you have to add the Pr of getting 70, 80, 90, 100
Pr(70:5 more right)=8C5 (.2)^5(.8)^3
Pr(80:6 right)=8C6 (.2)^6 (.8)^2
Pr(90:7 right)=8C7 (.2)^7 (.8)
Pr (100: 8 more right)= .2^8
To solve this problem using the binomial distribution, we need to calculate the probability of getting a specific number of correct answers out of the remaining eight questions that Carman guesses.
Carman knows the correct answers for questions 1 and 4, so she is guaranteed to get those two questions correct. This means she needs to get 60% of the remaining questions correct, which is 6 out of 8 questions.
Since each question has five possible answers and she is guessing randomly, the probability of guessing a single question correctly is 1/5 = 0.2. The probability of guessing incorrectly is 1 - 0.2 = 0.8.
Using the binomial distribution formula, we can calculate the probability of getting exactly 6 questions correct out of the remaining 8 questions:
P(X = 6) = C(8, 6) * (0.2)^6 * (0.8)^(8-6)
Where C(n, r) is the binomial coefficient, which represents the number of ways to choose r items out of n. It is calculated as:
C(n, r) = n! / (r!(n-r)!)
Applying this to our problem:
P(X = 6) = C(8, 6) * (0.2)^6 * (0.8)^(8-6)
= 28 * 0.2^6 * 0.8^2
Evaluating this expression will give you the probability of Carman getting exactly 6 questions correct out of the remaining 8 guesses.
Note: To calculate the overall probability of Carman getting 60% on the quiz, you would need to consider all possible combinations of getting exactly 6 questions correct out of different positions within the 8 remaining questions. This would involve summing up the probabilities of getting 6 correct answers in positions 1-6, 2-7, 3-8, and so on, up to position 8.