a quiz had 6 questions, each with 4 possible answers (only one of which is correct). If you answer them at random, find the probability that all are correct and at least one is wrong.
I assume you want two answers,since "all are correct and at least one is wrong" is impossible.
P(all correct) = 1/4^6
P(some wrong) = 1 - P(all correct)
To find the probability that all six answers are correct, we need to determine the probability of selecting the correct answer for each question and multiply them together.
Since each question has 4 possible answers and only one is correct, the probability of choosing the correct answer for each question is 1/4. Thus, the probability of all six answers being correct is (1/4) * (1/4) * (1/4) * (1/4) * (1/4) * (1/4) = 1/4096.
Next, to find the probability of at least one answer being wrong, we can subtract the probability of all answers being correct from 1.
Therefore, the probability of at least one answer being wrong is 1 - 1/4096 = 4095/4096.
So, the probability that all answers are correct is 1/4096, and the probability that at least one answer is wrong is 4095/4096.