suppose a student takes a multiple choice test with 10 questions. suppose each question has 4 possible answers. if the student guesses randomly on each questions,what is the probability that the student gets exactly 6 correct? less than 7 are correct?
To find the probability of a specific number of correct answers on a multiple-choice test, we can use the concept of probability and the binomial distribution formula.
The probability of getting exactly 6 correct answers can be calculated as follows:
1. Use the binomial distribution formula: P(x = k) = C(n, k) * p^k * (1-p)^(n-k), where:
- P(x = k) is the probability of getting exactly k correct answers,
- n is the total number of questions (10 in this case),
- k is the number of correct answers (6 in this case),
- p is the probability of getting a single question correct (1/4 since there are 4 possible answers).
2. Substitute the values into the formula:
P(x = 6) = C(10, 6) * (1/4)^6 * (1-(1/4))^(10-6).
C(10, 6) represents the number of ways to choose 6 questions out of 10, which can be calculated using the binomial coefficient formula: C(n, k) = n! / (k! * (n-k)!).
In this case, C(10, 6) = 10! / (6! * (10-6)!) = 210.
3. Calculate the probability:
P(x = 6) = 210 * (1/4)^6 * (3/4)^4.
The probability that the student gets exactly 6 questions correct is P(x = 6).
To calculate the probability that the student gets less than 7 questions correct, we need to sum the probabilities of getting exactly 0, 1, 2, 3, 4, 5, and 6 correct answers.
1. Calculate the probabilities for each case as explained above.
2. Sum the individual probabilities: P(x < 7) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6).
Note: If you'd like me to precisely calculate these probabilities for you, I'd be happy to do so!
To calculate the probability that the student gets exactly 6 correct out of 10 questions with 4 possible answers for each question, we will use the binomial probability formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
where:
P(X=k) is the probability of getting exactly k correct answers,
C(n, k) is the combination formula (n choose k),
p is the probability of guessing the correct answer (1/4 in this case),
n is the total number of questions (10 in this case), and
k is the number of correct answers (6 in this case).
1. Probability of getting exactly 6 correct answers:
P(X=6) = C(10, 6) * (1/4)^6 * (3/4)^(10-6)
Calculating the combination:
C(10, 6) = (10! / (6! * (10-6)! )
C(10, 6) = 210
Calculating the probability:
P(X=6) = 210 * (1/4)^6 * (3/4)^4
P(X=6) = 210 * (1/4096) * (81/256)
P(X=6) ≈ 0.200
Therefore, the probability that the student gets exactly 6 correct answers is approximately 0.200 or 20%.
2. Probability that less than 7 are correct:
To find the probability that less than 7 answers are correct, we need to calculate the probabilities of getting 0, 1, 2, 3, 4, 5, and 6 correct answers and then sum them up.
P(X<7) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)
Calculating each probability individually:
P(X=0) = C(10, 0) * (1/4)^0 * (3/4)^(10-0)
P(X=0) = 210 * 1 * (81/256)^10
P(X=1) = C(10, 1) * (1/4)^1 * (3/4)^(10-1)
P(X=1) = 210 * (1/4) * (3/4)^9
P(X=2) = C(10, 2) * (1/4)^2 * (3/4)^(10-2)
P(X=2) = 210 * (1/16) * (3/4)^8
P(X=3) = C(10, 3) * (1/4)^3 * (3/4)^(10-3)
P(X=3) = 210 * (1/64) * (3/4)^7
P(X=4) = C(10, 4) * (1/4)^4 * (3/4)^(10-4)
P(X=4) = 210 * (1/256) * (3/4)^6
P(X=5) = C(10, 5) * (1/4)^5 * (3/4)^(10-5)
P(X=5) = 210 * (1/1024) * (3/4)^5
P(X=6) = C(10, 6) * (1/4)^6 * (3/4)^(10-6)
P(X=6) = 210 * (1/4096) * (3/4)^4
Summing up the probabilities:
P(X<7) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)
P(X<7) ≈ 0.997
Therefore, the probability that the student gets less than 7 correct answers is approximately 0.997 or 99.7%.