A quiz consists of 10 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 40 %
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To get 40% or more, you cannot have
0, 1, 2, or 3 only correct answers
prob(0 right) = C(10,0) (1/5)^0 (4/5)^10 = .10737
prob(1 right) = C(10,1)(1/5)(4/5)^9 = .26844
prob(2right) = C(10,2)(1/5)^2(4/5)^8 = .30199
prob(3right) = C(10,3)(1/5)3 (4/5)^7 = . 20133
total = .87913
so prob of passing by just guessing = 1 - .87913 = .1209
check my arithmetic, the procedure is correct.
To find the probability of passing the quiz by random guessing, we need to calculate the probability of getting at least 40% of the questions correct out of the 10 multiple-choice questions.
To pass the quiz with a minimum of 40%, a student needs to get a minimum of 4 questions correct (0.4 * 10 = 4). Anything less than 4 correct answers would result in a grade lower than 40%.
The probability of getting exactly 4 questions correct is calculated as:
P(4 correct) = (number of ways to get 4 correct) * (probability of getting each question correct) * (probability of getting each question incorrect for the remaining 6 questions)
The number of ways to get 4 correct answers out of 10 is calculated using the binomial coefficient formula:
(number of ways to choose 4 out of 10) = 10! / (4! * (10 - 4)!)
The probability of getting each question correct is 1 / 5 since there are 5 possible answers for each question and only one of them is correct.
The probability of getting each question incorrect for the remaining 6 questions is (4 / 5) ^ 6 since there are 4 incorrect answers out of 5 possible answers for each question, and we have 6 remaining questions.
Now we can calculate the probability of passing the quiz:
P(passing) = P(4 correct) + P(5 correct) + ... + P(10 correct)
P(passing) = (10! / (4! * 6!)) * (1 / 5) ^ 4 * (4 / 5) ^ 6 + (10! / (5! * 5!)) * (1 / 5) ^ 5 * (4 / 5) ^ 5 + ... + (10! / (10! * 0!)) * (1 / 5) ^ 10 * (4 / 5) ^ 0
Now you can calculate this expression to get the probability of passing the quiz by random guessing.
To find the probability of passing the quiz by making random guesses, we first need to determine the minimum number of correct answers needed to achieve a passing grade of 40%.
1. Calculate the minimum number of correct answers needed:
10 questions * 40% passing grade = 4 correct answers needed
2. Find the probability of getting exactly 4 correct answers by making random guesses:
Each question has 5 possible answers, so the probability of guessing the correct answer for each question is 1/5.
However, there are 10 questions, so the probability of getting exactly 4 correct answers is calculated using the binomial probability formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
where P(X = k) is the probability of getting exactly k correct answers, n is the number of questions, k is the number of correct answers needed, p is the probability of getting a question correct (1/5), and nCk is the binomial coefficient (n choose k).
Plugging in the values:
P(X = 4) = 10C4 * (1/5)^4 * (4/5)^(10-4)
3. Calculate the probability of getting 4 or more correct answers:
To find the probability of passing, we need to calculate the probability of getting 4, 5, 6, 7, 8, 9, or 10 correct answers. This can be done by summing the probabilities of getting each of these individual numbers of correct answers.
4. Calculate the cumulative probability of passing:
The cumulative probability of passing the quiz is the probability of getting 4 or more correct answers.
P(Passing) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
Therefore, the final answer depends on calculating these probabilities using the binomial probability formula.