A student takes a 32 question multiple choice exam but did not study and randomly guesses each answer. Each question has three possible choices for the answer. Find the probability that the student guesses more than 75% of the questions correctly. (Round your answer to three decimal places.)
To find the probability that the student guesses more than 75% of the questions correctly, we need to calculate the probability of getting each possible percentage correct.
Since each question has three possible choices for the answer, the probability of guessing any one question correctly is 1/3.
Let's define X as the number of questions the student guesses correctly out of the 32 total questions.
Now, let's calculate the probability of guessing each possible percentage of questions correctly:
- To guess 75% of the questions correctly, the student needs to answer 24 correct and 8 incorrect. The probability of guessing 24 questions correctly and 8 questions incorrectly is given by the binomial probability formula:
P(X=24) = C(32, 24) * (1/3)^24 * (2/3)^8
- To guess 76% of the questions correctly, the student needs to answer 25 correct and 7 incorrect. The probability of guessing 25 questions correctly and 7 questions incorrectly is given by:
P(X=25) = C(32, 25) * (1/3)^25 * (2/3)^7
Similarly, we can calculate the probabilities for guessing 77% and above:
P(X=26), P(X=27), ..., P(X=32)
Finally, to find the probability that the student guesses more than 75% of the questions correctly, we sum up the probabilities for guessing 76%, 77%, ..., 100%:
P[X > 75%] = P(X=25) + P(X=26) + ... + P(X=32)
Calculating these probabilities and rounding to three decimal places, we get the answer.