If a student randomly guesses at 15 multiple-choice questions, find the probability that the student gets exactly seven correct. Each question has four possible choices
for any question,
prob(correct guess) = 1/4
prob(wrong guess) = 3/4
To have exactly 7 of 15 correct,
prob = C(15,7)((1/4)^7 (3/4)^8
= 6435 (1/16384)(6561/65536)
= appr .039
If a student randomly guesses at 14 multiple-choice questions, find the probability that the student gets exactly five correct. Each question has five possible choices.
If a student randomly guesses at 15 multiple-choice questions, find the probability that the student gets exactly four correct. Each question has four possible choices.
To find the probability that the student gets exactly seven correct out of 15 multiple-choice questions, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the total number of trials
- k is the number of successes
- p is the probability of success in a single trial
- (1-p) is the probability of failure in a single trial
- C(n, k) is the number of ways to choose k successes from n trials, also known as the binomial coefficient
In this case, we have:
- n = 15 (total number of questions)
- k = 7 (number of correct answers)
- p = 1/4 (probability of choosing the correct answer in a single trial, since there are four possible choices)
To calculate the binomial coefficient, we use the formula:
C(n, k) = n! / (k!(n-k)!)
where ! denotes factorial.
Therefore, to find the probability:
P(X = 7) = C(15, 7) * (1/4)^7 * (3/4)^(15-7)
Now, let's calculate each component:
1. C(15, 7):
= 15! / (7! * (15-7)!)
= (15 * 14 * 13 * 12 * 11 * 10 * 9) / (7 * 6 * 5 * 4 * 3 * 2 * 1)
= 6435
2. (1/4)^7:
= (1/4) * (1/4) * (1/4) * (1/4) * (1/4) * (1/4) * (1/4)
= 1/16384
3. (3/4)^(15-7):
= (3/4)^8
= 6561/65536
Finally, we can calculate the probability:
P(X = 7) = C(15, 7) * (1/4)^7 * (3/4)^(15-7)
= 6435 * (1/16384) * (6561/65536)
≈ 0.2508
Therefore, the probability that the student gets exactly seven correct out of 15 multiple-choice questions is approximately 0.2508, or 25.08%.