A multiple-choice quiz has 15 questions, each with 4 possible answers of which only 1 is the correct answer. What is the probability that a student gets 7 to 11 correct answer?
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To find the probability that a student gets 7 to 11 correct answers in a multiple-choice quiz with 15 questions, we need to consider the number of ways the student can choose 7, 8, 9, 10, or 11 correct answers out of the 15 questions.
The probability of getting a single question correct is 1/4, and the probability of getting a single question wrong is 3/4.
Let's calculate the probability for each possible number of correct answers and then sum them up.
For getting exactly 7 correct answers:
P(7 correct) = (15 choose 7) * (1/4)^7 * (3/4)^(15-7)
For getting exactly 8 correct answers:
P(8 correct) = (15 choose 8) * (1/4)^8 * (3/4)^(15-8)
For getting exactly 9 correct answers:
P(9 correct) = (15 choose 9) * (1/4)^9 * (3/4)^(15-9)
For getting exactly 10 correct answers:
P(10 correct) = (15 choose 10) * (1/4)^10 * (3/4)^(15-10)
For getting exactly 11 correct answers:
P(11 correct) = (15 choose 11) * (1/4)^11 * (3/4)^(15-11)
Now we can sum these probabilities to get the total probability of getting 7 to 11 correct answers:
P(7 to 11 correct) = P(7 correct) + P(8 correct) + P(9 correct) + P(10 correct) + P(11 correct)
You can calculate the probabilities using the binomial coefficient formula:
(n choose k) = n! / ((n-k)! * k!)
Where "n" is the total number of questions (15) and "k" is the number of correct answers.
Let's calculate the probabilities step-by-step.
To find the probability that a student gets 7 to 11 correct answers on the multiple-choice quiz, we need to calculate the sum of the probabilities of getting 7, 8, 9, 10, or 11 questions correct.
Let's break it down step by step:
Step 1: Find the probability of getting exactly k questions correct, where k varies from 7 to 11.
To calculate the probability of getting exactly k questions correct, we'll use the binomial probability formula:
P(k successes) = C(n, k) * p^k * q^(n-k)
where:
- P(k successes) is the probability of getting exactly k questions correct,
- C(n, k) is the number of ways to choose k questions out of n questions (given by the binomial coefficient),
- p is the probability of getting a question correct (1/4 in this case),
- q is the probability of getting a question wrong (3/4 in this case), and
- n is the total number of questions (15 in this case).
Step 2: Calculate the probabilities for k = 7, 8, 9, 10, and 11.
Using the binomial probability formula, calculate the probability for each value of k:
P(7) = C(15, 7) * (1/4)^7 * (3/4)^(15-7)
P(8) = C(15, 8) * (1/4)^8 * (3/4)^(15-8)
P(9) = C(15, 9) * (1/4)^9 * (3/4)^(15-9)
P(10) = C(15, 10) * (1/4)^10 * (3/4)^(15-10)
P(11) = C(15, 11) * (1/4)^11 * (3/4)^(15-11)
Step 3: Sum up the probabilities.
Add up the probabilities for k = 7, 8, 9, 10, and 11 to get the probability of getting 7 to 11 correct answers:
P(7 to 11) = P(7) + P(8) + P(9) + P(10) + P(11)
Calculate each individual probability using the formula, and then add them together to find the final probability.