A student takes an exam containing 14 multiple choice questions. The probability of choosing a correct answer by knowledgeable guessing is 0.5. At least 9 correct answers are required to pass. If the student makes knowledgeable guesses, what is the probability that he will fail? Round your answer to four decimal places.
0.2120
To find the probability that the student will fail, we need to calculate the probability of getting fewer than 9 correct answers.
The probability of getting exactly k correct answers out of n questions using knowledgeable guessing is given by the binomial distribution formula:
P(X=k) = (n choose k) * p^k * q^(n-k)
Where:
- n is the total number of questions (14 in this case).
- k is the number of correct answers (less than 9 in this case).
- p is the probability of getting a correct answer (0.5 in this case, since it's a knowledgeable guess).
- q is the probability of getting an incorrect answer (1 - p = 0.5).
Now let's calculate the probability of getting fewer than 9 correct answers by summing up the probabilities for k=0 to k=8.
P(fail) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=8)
= Σ(P(X=k)), for k=0 to k=8
Let's calculate this probability step-by-step:
1. Calculate P(X=k) for each value of k from 0 to 8:
P(X=0) = (14 choose 0) * 0.5^0 * 0.5^14 = 1 * 1 * 0.0000190734863281 ≈ 0.0000191
P(X=1) = (14 choose 1) * 0.5^1 * 0.5^13 = 14 * 0.5 * 0.00003814697265625 ≈ 0.0005321
P(X=2) = (14 choose 2) * 0.5^2 * 0.5^12 = 91 * 0.25 * 0.0000762939453125 ≈ 0.0011445
P(X=3) = (14 choose 3) * 0.5^3 * 0.5^11 = 364 * 0.125 * 0.000152587890625 ≈ 0.0034332
P(X=4) = (14 choose 4) * 0.5^4 * 0.5^10 = 1001 * 0.0625 * 0.00030517578125 ≈ 0.0072575
P(X=5) = (14 choose 5) * 0.5^5 * 0.5^9 = 2002 * 0.03125 * 0.0006103515625 ≈ 0.0117645
P(X=6) = (14 choose 6) * 0.5^6 * 0.5^8 = 3003 * 0.015625 * 0.001220703125 ≈ 0.0169353
P(X=7) = (14 choose 7) * 0.5^7 * 0.5^7 = 3432 * 0.0078125 * 0.00244140625 ≈ 0.0177813
P(X=8) = (14 choose 8) * 0.5^8 * 0.5^6 = 3003 * 0.00390625 * 0.0048828125 ≈ 0.0115166
2. Sum up the probabilities for k=0 to k=8:
P(fail) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=8)
= 0.0000191 + 0.0005321 + 0.0011445 + 0.0034332 + 0.0072575 + 0.0117645 + 0.0169353 + 0.0177813 + 0.0115166
3. Round the result to four decimal places:
P(fail) ≈ 0.0719
Therefore, the probability that the student will fail is approximately 0.0719.
To find the probability that the student will fail, we need to find the probability of getting less than 9 correct answers when making knowledgeable guesses.
The probability of getting a question correct by knowledgeable guessing is 0.5, and the probability of getting a question wrong is also 0.5.
To find the probability of getting exactly k correct answers out of 14 questions, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- n is the total number of trials (in this case, the number of questions)
- k is the number of successful outcomes (in this case, the number of correct answers)
- p is the probability of success (in this case, 0.5)
Since we need to calculate the probability of getting less than 9 correct answers, we need to sum the individual probabilities for k = 0, 1, 2, 3, 4, 5, 6, 7, and 8 (since 9 or above would be a passing grade).
P(fail) = P(X < 9) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
Let's calculate this using the formula: