It is a multiple choice exam with 12 problems, each problem has 5 options. You need to answer 10 correctly to pass, what is the probability that by guessing you will answer at least 10 problems correctly?
at least 10? that means you can get 2 or less wrong.
Pr(2 wrong)=(4/5)^2*(1/5)^10
pr(1 wrong)=(4/5)(1/5)^11
Pr(0 wrong)=(1/5)^12
so now, Pr( at least 10 right)=1-pr(2wrong)-pr(1 wrong)-pr(zero wrong)
To calculate the probability of answering at least 10 problems correctly by guessing, we need to consider the number of ways to choose 10, 11, or 12 correct answers out of 12 questions.
First, let's calculate the probability of guessing a single problem correctly. Since each problem has 5 options, the probability of guessing one problem correctly is 1/5.
Now, let's calculate the probability of guessing exactly 10, 11, or 12 problems correctly. We'll use the binomial probability formula:
P(X=k) = (nCk) * 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 (12 problems)
- k is the number of successful trials (10, 11, or 12 problems)
- p is the probability of a single success (1/5)
- (nCk) is the number of combinations (12Ck) of choosing k problems correctly out of 12
To calculate (nCk), we can use the formula for combinations:
(nCk) = n! / (k!(n-k)!)
Let's calculate the probabilities:
For k = 10:
P(X=10) = (12C10) * (1/5)^10 * (4/5)^2
For k = 11:
P(X=11) = (12C11) * (1/5)^11 * (4/5)^1
For k = 12:
P(X=12) = (12C12) * (1/5)^12 * (4/5)^0
Finally, we can sum these probabilities to get the probability of answering at least 10 problems correctly:
P(X>=10) = P(X=10) + P(X=11) + P(X=12)
Note: (12C10) = (12C2) = 66, (12C11) = (12C1) = 12, (12C12) = 1
Calculating the probabilities, we find:
P(X=10) = 66 * (1/5)^10 * (4/5)^2 ≈ 0.0263
P(X=11) = 12 * (1/5)^11 * (4/5)^1 ≈ 0.0017
P(X=12) = 1 * (1/5)^12 * (4/5)^0 ≈ 0.00003
P(X>=10) = P(X=10) + P(X=11) + P(X=12) ≈ 0.0280
Therefore, the probability of answering at least 10 problems correctly by guessing is approximately 0.0280 or 2.80%.