A quiz consists of 20 multiple-choice questions, each with 5 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 70 %.
To get 70% or more of 20 questions means that you must get at least 14 of them correct.
prob(correct) = 1/5, prob(not correct) = 4/5
prob(pass)
= C(20,14) (.2)^14 (.8)^6 + C(20,15) (.2)^15 (.8)^5 + ...
... + C(20,20)(.2)^20 (.8)^0
= .000001664 + .000000166 + ..... + 1.0486(10^-14)
= appr .00000183
since the terms get rapidly smaller, I only added the first two terms
(my calculator kicked into scientific notation of the answer after the 3rd term)
To pass the quiz, one must answer at least 14 out of the 20 questions correctly. Since each question has 5 possible answers and someone is making random guesses, the probability of getting any single question correct is 1/5 or 0.2.
The probability of getting exactly 14 questions correct can be calculated using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the total number of trials (20 questions)
- k is the number of successful trials (14 correct answers)
- p is the probability of a successful trial (0.2)
- (n choose k) represents the binomial coefficient, which calculates the number of ways to choose k successes from n trials
P(X=14) = (20 choose 14) * (0.2)^14 * (1-0.2)^(20-14)
Using a calculator or software, we can calculate:
(20 choose 14) = 38760
P(X=14) = 38760 * (0.2)^14 * (0.8)^6 = 0.0256
Now, we need to calculate the probability of getting exactly 15, 16, 17, 18, 19, or 20 questions correct. Since these probabilities are calculated in the same way as the previous one, we can use the same formulas.
P(X=15) = (20 choose 15) * (0.2)^15 * (1-0.2)^(20-15) = 0.0036
P(X=16) = (20 choose 16) * (0.2)^16 * (1-0.2)^(20-16) = 0.0005
P(X=17) = (20 choose 17) * (0.2)^17 * (1-0.2)^(20-17) = 0.00006
P(X=18) = (20 choose 18) * (0.2)^18 * (1-0.2)^(20-18) = 0.000006
P(X=19) = (20 choose 19) * (0.2)^19 * (1-0.2)^(20-19) = 0.0000003
P(X=20) = (20 choose 20) * (0.2)^20 * (1-0.2)^(20-20) = 0.00000001
Finally, we need to add up these probabilities to get the overall probability of passing:
P(passing) = P(X >= 14) = P(X=14) + P(X=15) + P(X=16) + P(X=17) + P(X=18) + P(X=19) + P(X=20)
P(passing) = 0.0256 + 0.0036 + 0.0005 + 0.00006 + 0.000006 + 0.0000003 + 0.00000001
P(passing) = 0.029776
To find the probability of passing the quiz by making random guesses, we need to determine the minimum number of correct answers needed to achieve a passing grade of 70%.
First, let's calculate the minimum number of correct answers needed to pass the quiz. Since there are 20 questions in total, multiplying 20 by 0.7 gives us 14. Thus, you need to answer at least 14 questions correctly to pass.
Now, let's calculate the probability of randomly guessing the correct answer for each question. Since each question has 5 possible answers, the probability of guessing the correct answer for each question is 1 out of 5, or 1/5.
To determine the probability of answering at least 14 questions correctly, we can use the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the total number of trials (which is 20 in this case)
- k is the desired number of successful trials (which is 14 in this case)
- p is the probability of success (which is 1/5 in this case)
Now, let's calculate the probability of passing the quiz:
P(X >= 14) = P(X = 14) + P(X = 15) + ... + P(X = 20)
P(X >= 14) = (20 Choose 14) * (1/5)^14 * (4/5)^(20-14) + (20 Choose 15) * (1/5)^15 * (4/5)^(20-15) + ... + (20 Choose 20) * (1/5)^20 * (4/5)^(20-20)
Using a calculator or a statistical software, evaluate the sum of these probabilities to find the final probability of passing the quiz by making random guesses.