A student takes a multiple choice quiz with 12 questions and 5 alternatives per question. Unfortunately, this student has no idea what any of the answers are, so he randomly selects an alternative for each question on the exam. Using this “blind guessing” strategy, how likely is the student to get:
(a). exactly 5 answers correct?
(b). exactly 2 answers correct?
(c). 4 or fewer answers correct?
(d). He wanted to get at least 75% on the quiz. How likely is that to happen?
prob(correct) = 1/5
prob(not correct) = 4/5
a) to have exactly 5 correct
= C(12,5) (1/5)^5 (4/5)^7
= appr .05315
b) prob(2 correct)
= C(12,2) (1/5)^2 (4/5)^10
= ...
c) 4 or fewer correct
---> 0 correct + 1 correct + 2 correct + 3 correct + 4 correct
d) to get 75% he needs to have at least 9 correct
so find
9 correct + 10 correct + 11 correct + 12 correct
lots of calculations, follow the method I used in a) and b)
To find the probabilities for each case, we need to use a method called the binomial probability formula. In this case, the formula is:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where P(X=k) is the probability of getting exactly k answers correct, C(n,k) is the number of different ways to choose k correct answers out of n total questions, p is the probability of getting a single question correct (1/5 in this case), and (1-p) is the probability of getting a single question wrong (4/5 in this case).
Now let's solve each case:
(a). Exactly 5 answers correct: P(X=5)
Using the binomial probability formula:
P(X=5) = C(12,5) * (1/5)^5 * (4/5)^7 ≈ 0.073
(b). Exactly 2 answers correct: P(X=2)
Using the binomial probability formula:
P(X=2) = C(12,2) * (1/5)^2 * (4/5)^10 ≈ 0.120
(c). 4 or fewer answers correct: P(X ≤ 4)
To find this probability, we need to sum up the probabilities for each case from 0 to 4:
P(X ≤ 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)
P(X ≤ 4) = C(12,0) * (1/5)^0 * (4/5)^12 + C(12,1) * (1/5)^1 * (4/5)^11 + C(12,2) * (1/5)^2 * (4/5)^10 + C(12,3) * (1/5)^3 * (4/5)^9 + C(12,4) * (1/5)^4 * (4/5)^8
P(X ≤ 4) ≈ 0.891
(d). At least 75% of the quiz correct: P(X ≥ 0.75*12)
To find this probability, we need to sum up the probabilities for each case from 9 to 12:
P(X ≥ 0.75*12) = P(X=9) + P(X=10) + P(X=11) + P(X=12)
P(X ≥ 0.75*12) = C(12,9) * (1/5)^9 * (4/5)^3 + C(12,10) * (1/5)^10 * (4/5)^2 + C(12,11) * (1/5)^11 * (4/5)^1 + C(12,12) * (1/5)^12 * (4/5)^0
P(X ≥ 0.75*12) ≈ 0.005
So, the probabilities are:
(a). P(X=5) ≈ 0.073
(b). P(X=2) ≈ 0.120
(c). P(X ≤ 4) ≈ 0.891
(d). P(X ≥ 0.75*12) ≈ 0.005
To find the probability of getting a certain number of answers correct by blind guessing, we can use the concept of binomial probability. The formula for binomial probability is:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting k successes
n is the total number of trials
k is the number of desired successes
p is the probability of success on a single trial (in this case, the chance of guessing the correct answer, which is 1/5)
(1-p) is the probability of failure on a single trial
Now, let's calculate the probabilities for each of the given questions:
(a) To find the probability of exactly 5 answers correct:
P(X=5) = (12 choose 5) * (1/5)^5 * (4/5)^(12-5) = 792 * (1/5)^5 * (4/5)^7 ≈ 0.0262 or 2.62%
(b) To find the probability of exactly 2 answers correct:
P(X=2) = (12 choose 2) * (1/5)^2 * (4/5)^(12-2) = 66 * (1/5)^2 * (4/5)^10 ≈ 0.2907 or 29.07%
(c) To find the probability of getting 4 or fewer answers correct, we need to calculate the sum of probabilities for each individual case:
P(X≤4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)
= (12 choose 0) * (1/5)^0 * (4/5)^12 + (12 choose 1) * (1/5)^1 * (4/5)^11 + (12 choose 2) * (1/5)^2 * (4/5)^10 + (12 choose 3) * (1/5)^3 * (4/5)^9 + (12 choose 4) * (1/5)^4 * (4/5)^8
≈ 0.2305 or 23.05%
(d) To find the probability of getting at least 75% (at least 9 out of 12) correct, we need to calculate the sum of probabilities for each individual case:
P(X≥9) = P(X=9) + P(X=10) + P(X=11) + P(X=12)
= (12 choose 9) * (1/5)^9 * (4/5)^(12-9) + (12 choose 10) * (1/5)^10 * (4/5)^(12-10) + (12 choose 11) * (1/5)^11 * (4/5)^(12-11) + (12 choose 12) * (1/5)^12 * (4/5)^(12-12)
≈ 0.0496 or 4.96%
So, the probabilities are:
(a) The probability of exactly 5 answers correct is approximately 2.62%.
(b) The probability of exactly 2 answers correct is approximately 29.07%.
(c) The probability of getting 4 or fewer answers correct is approximately 23.05%.
(d) The probability of getting at least 75% correct is approximately 4.96%.