A quiz has 20 multiple choice questions, each with a choice of 4 answers. If Jay guesses an answer for each question at random, find the probability that he will score 75% on this quiz, correct to 2 significant figures.
Answer: 0.000 003 4
Thanks for any help
this is a binomial probability
... p(correct) = .25
... p(wrong) = .75
(c + w)^20
p(75% correct) = 20C5 * .25^15 * .75^5
To find the probability that Jay will score 75% on the quiz, we need to calculate the probability of answering 15 questions correctly out of 20.
The probability of Jay guessing the correct answer for each question is 1/4, since there are 4 choices for each question. Therefore, the probability of answering one specific question correctly is 1/4.
To calculate the probability of answering 15 questions correctly, we need to consider the number of ways Jay can choose 15 questions out of 20 to answer correctly, multiplied by the probability of answering these 15 questions correctly, multiplied by the probability of answering the remaining 5 questions incorrectly.
The number of ways to choose 15 questions out of 20 can be calculated using the binomial coefficient formula, which is found by dividing the factorial of the total number of questions (20!) by the product of the factorial of the desired number of correct answers (15!) and the factorial of the remaining questions (20 - 15)!.
The probability of answering 15 questions correctly is (1/4)^15 since each question is independent.
The probability of answering the remaining 5 questions incorrectly is (3/4)^5 since each question has a 3/4 chance of being answered incorrectly (choosing the wrong answer out of the 4 choices).
Putting all these calculations together, we can find the probability of scoring 75% on the quiz:
Probability = (number of ways to choose 15 questions out of 20) * (probability of answering 15 questions correctly) * (probability of answering the remaining 5 questions incorrectly)
Probability = (20! / (15! * (20 - 15)!) * (1/4)^15 * (3/4)^5
Calculating this expression will give us the desired probability. Rounding to 2 significant figures, the answer is approximately 0.0000034 (or 3.4 x 10^-6)