A multiple choice test has 12 questions each of which has 5 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly?
Solution:
A multiple choice test has 12 questions each of which has 5 possible answers, only one of which is correct.
Judy, who forgot to study for the test, guesses on all questions.
We have to find the probability that she will answer exactly 3 questions correctly.
Total number of options = 12(5) = 60
Total number of correct answers = 12
Total number of wrong answers = 60 - 12 = 48
Probability of choosing correct answer = 12/60
= 1/5
= 0.2
Probability of choosing wrong answers = 48/60
= 4/5
= 0.8
By using binomial distribution,
\(\sum_{k=0}^{n}\, ^{n}C_{k}(p)^{k}(q)^{(n-k)}\)
Where, p is the probability of success every trial
q is the probability of failure every trial
n is the total number of trials
k is the total number of target successes
Here, n = 12, k = 3, p = 0.2 and q = 0.8
So, \(\sum_{3}^{12}\, ^{12}C_{3}(0.2)^{3}(0.8)^{(12-3)} = (\frac{12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{(3\times 2\times 1)(9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1)})(0.2)^{3} (0.4)^{9}\)
\(= (\frac{12\times 11\times 10}{3\times 2\times 1})(0.2)^{3} (0.8)^{9}\)
= (120×11/6)(0.008)(0.134217728)
= 0.236
Therefore, the probability of answering exactly 3 questions correctly is 0.236.
A multiple choice test has 12 questions each of which has 5 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly?
Summary:
A multiple choice test has 12 questions each of which has 5 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, the probability that she will answer exactly 3 questions correctly is 0.236
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