An unprepared student makes random guesses for the ten true-false questions on a quiz. Find the probability that there is at least one correct answer.
0.999
The probability for at least one correct answer, P(X>=1) is the same as probability of not getting ALL wrong answers or 1-P(X=0).
P(X=0) is getting every one wrong, with a probability of (1/2)^n, where n is the number of questions guessed.
To find the probability that there is at least one correct answer, we need to calculate the probability of the complement event, i.e., the event of getting no correct answers, and then subtract it from 1.
First, let's determine the probability of getting one question correct. Since there are two options (true or false) for each question, the probability of guessing the correct answer to a single true-false question is 1/2.
The probability of not getting any questions correct on a single guess is the complement of getting one question correct, which is 1 - 1/2 = 1/2.
Now, let's determine the probability of not getting any correct answers on all ten questions. Since the student is making random guesses for each question independently, the probability of not getting any correct answers on all ten questions is simply (1/2)^10.
Finally, we can calculate the probability of getting at least one correct answer by subtracting the probability of no correct answers from 1:
P(at least one correct answer) = 1 - P(no correct answers)
= 1 - (1/2)^10
= 1 - 1/1024
= 1023/1024
Therefore, the probability that there is at least one correct answer is 1023/1024.