A student is taking a multiple-choice exam with 16 questions. Each question has five alternatives. If the student guesses on 12 of the questions, what is the probability she will guess at least 8 correct? Assume all of the alternatives are equally likely for each question on which the student guesses.
I thought this was pretty easy but I cannot figure it out!
You need to add the probabilities of getting 8,9,10,11,12,13,14,15 and 16 right
The probability of getting 8 right (and 8 wrong) is
(0.2)^8*(0.8)^8*[16!/(8!*8!)]= 0.00553
The probability of getting 9 right (and 7 wrong) is
(0.2)^9*(0.8)^7*[16!/(7!*9!)]= 0.00122
etc. The probability of getting higher numbers right drops off very fast
The probability of getting 16 right is (0.2)^16 = 7*10^-12 (i.e negligible)
.0069
To find the probability that the student will guess at least 8 questions correctly, we need to calculate the probability of guessing exactly 8, 9, 10, 11, or 12 questions correctly, and then sum up those probabilities.
Let's break it down step-by-step:
Step 1: Calculate the probability of guessing exactly 8 questions correctly.
The probability of guessing one question correctly is 1/5, and the probability of guessing one question incorrectly is 4/5. Since the student is guessing on all 12 questions, the probability of getting exactly 8 questions correct is calculated as follows:
P(8 correct) = (1/5)^8 * (4/5)^4 * (12 choose 8)
Step 2: Calculate the probability of guessing exactly 9, 10, 11, and 12 questions correctly.
Using the same logic as in step 1, we can calculate the probabilities for each scenario:
P(9 correct) = (1/5)^9 * (4/5)^3 * (12 choose 9)
P(10 correct) = (1/5)^10 * (4/5)^2 * (12 choose 10)
P(11 correct) = (1/5)^11 * (4/5)^1 * (12 choose 11)
P(12 correct) = (1/5)^12 * (4/5)^0 * (12 choose 12)
Step 3: Sum up the probabilities.
Finally, to get the probability of guessing at least 8 questions correctly, we sum up the probabilities from steps 1 and 2:
P(at least 8 correct) = P(8 correct) + P(9 correct) + P(10 correct) + P(11 correct) + P(12 correct)
Let's calculate the probabilities:
P(8 correct) = (1/5)^8 * (4/5)^4 * (12 choose 8)
= 0.0138889
P(9 correct) = (1/5)^9 * (4/5)^3 * (12 choose 9)
= 0.002315
P(10 correct) = (1/5)^10 * (4/5)^2 * (12 choose 10)
= 0.0002315
P(11 correct) = (1/5)^11 * (4/5)^1 * (12 choose 11)
= 0.000012157
P(12 correct) = (1/5)^12 * (4/5)^0 * (12 choose 12)
= 5.505e-07
P(at least 8 correct) = 0.0138889 + 0.002315 + 0.0002315 + 0.000012157 + 5.505e-07
≈ 0.01644
Therefore, the probability that the student will guess at least 8 questions correctly is approximately 0.01644 or 1.644%.
To determine the probability of the student guessing at least 8 questions correctly, we can 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 at least k successes,
- C(n, k) is the number of combinations of n items taken k at a time (n choose k),
- p is the probability of success on a single trial, and
- n is the number of trials.
In this case, since the student is guessing on 12 questions, the total number of trials (n) is 12. The probability of getting a question correct by guessing is 1/5 or 0.2 (since there are five alternatives). The number of total questions (k) is the range of successes we are interested in (8, 9, 10, 11, and 12).
Now we can calculate the probability for each value of k separately and then add them together to get the probability of getting at least 8 questions correct.
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
Let's calculate each term one by one:
For k = 8:
P(X = 8) = C(12, 8) * (0.2)^8 * (0.8)^4
For k = 9:
P(X = 9) = C(12, 9) * (0.2)^9 * (0.8)^3
For k = 10:
P(X = 10) = C(12, 10) * (0.2)^10 * (0.8)^2
For k = 11:
P(X = 11) = C(12, 11) * (0.2)^11 * (0.8)^1
For k = 12:
P(X = 12) = C(12, 12) * (0.2)^12 * (0.8)^0
Finally, add all these probabilities together to get the probability of guessing at least 8 questions correct:
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
I hope this explanation helps you understand the problem and how to calculate the probability.