A test consists of 10 truefalse questions, To pass the tost a student must answer at least 6 questions correctly, If a student guesses on each question, what is the probability that the student will pass the test? Round to three decimal places. A. 0.377 B. 0.172 C. 0.828 D. 0.205

To find the probability that a student will pass the test by guessing on each question, we need to calculate the probability of getting at least 6 correct answers out of 10 questions. The probability of getting a correct answer by guessing is \( \frac{1}{2} \) since there are two possible outcomes (true or false) and each outcome has an equal probability. The probability of getting at least 6 correct answers can be calculated using the binomial probability formula: \[ P(X \geq 6) = \sum_{k=6}^{10} \binom{10}{k} \left(\frac{1}{2}\right)^k \left(\frac{1}{2}\right)^{10-k} \] where: - \( P(X \geq 6) \) is the probability of getting at least 6 correct answers - \( \binom{10}{k} \) is the number of combinations of 10 questions taken k at a time - \( \left(\frac{1}{2}\right)^k \) is the probability of getting k correct answers - \( \left(\frac{1}{2}\right)^{10-k} \) is the probability of getting 10-k correct answers Let's calculate this probability. Find the sum by following steps: - step0: Solution: \(\sum _{k=6}^{10} { }_{10}C_{k}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{2}\right)^{10-k}\) - step1: Expand the expression: \(\sum _{k=6}^{10}\frac{10!}{k!\times \left(10-k\right)!}\times \left(\frac{1}{2}\right)^{k}\left(\frac{1}{2}\right)^{10-k}\) - step2: Calculate the value: \(\sum _{k=6}^{10}\frac{3628800}{k!\times \left(10-k\right)!}\times \left(\frac{1}{2}\right)^{k}\left(\frac{1}{2}\right)^{10-k}\) - step3: Multiply: \(\sum _{k=6}^{10}\frac{14175}{2^{2}\times k!\times \left(10-k\right)!}\) - step4: Rewrite the expression: \(\sum _{k=6}^{10}2^{-2}\left(k!\right)^{-1}\left(\left(10-k\right)!\right)^{-1}\times 14175\) - step5: Convert the expressions: \(2^{-2}\left(6!\right)^{-1}\left(\left(10-6\right)!\right)^{-1}\times 14175+2^{-2}\left(7!\right)^{-1}\left(\left(10-7\right)!\right)^{-1}\times 14175+2^{-2}\left(8!\right)^{-1}\left(\left(10-8\right)!\right)^{-1}\times 14175+2^{-2}\left(9!\right)^{-1}\left(\left(10-9\right)!\right)^{-1}\times 14175+2^{-2}\left(10!\right)^{-1}\left(\left(10-10\right)!\right)^{-1}\times 14175\) - step6: Calculate: \(\frac{105}{512}+\frac{15}{128}+\frac{45}{1024}+\frac{5}{512}+\frac{1}{1024}\) - step7: Calculate: \(\frac{193}{512}\) The probability that the student will pass the test by guessing on each question is approximately 0.377. Therefore, the correct answer is A. 0.377.