A test has 15 true or false questions and 10 multiple choice questions (each with 4 choices). All questions are weighted equally. Jack has not studied at all for the test and randomly guesses.
1. What is the probability that Jack gets 96% on the test?
a. 0.000000001
b. 0.000000002
c. 0.0000007
d. 0.000000004
A
2. Assuming that jack takes the same test 20 different times and the tests determine his final mark (weighted evenly); what is the probability that he will pass the course?
a. 0.8186
b. 0.2567
c. 0.7433
d. 0.1814
D
But Why?
To find the probability that Jack gets a specific score on the test, we need to consider the number of ways he can answer the questions correctly.
For the first question:
1. The probability of getting a true or false question correct by random guessing is 1/2, as there are two options (true or false).
2. The probability of getting a multiple choice question correct by random guessing is 1/4, as there are four options.
3. To get a 96% score on the test, Jack needs to answer 24 out of 25 questions correctly.
4. Out of the 25 questions, 15 are true or false and 10 are multiple choice.
5. The probability of answering exactly 24 questions correctly is obtained by multiplying the probability of getting 24 true or false questions correct (1/2)^24, and the probability of getting 0 or 1 multiple choice question correct ((1/4)^0 * 3/4^10), as he would need to answer all multiple choice questions incorrectly to total 24 correct answers.
6. Finally, we multiply the probabilities of getting the true or false questions and the multiple-choice question correct to get the overall probability of achieving a 96% score.
For the second question:
1. Since this question assumes Jack takes the same test 20 times, we need to find the probability of passing the course based on his average score over the 20 tests.
2. To pass the course, Jack needs to achieve a score of at least 60%, which is 75 out of 125 possible points (25 questions x 5 points each).
3. To find the probability of passing, we need to calculate the probability of Jack getting 75 or more questions correct on each of the 20 tests.
4. Using similar calculations as in the first question, we multiply the probabilities of getting the true or false questions and the multiple-choice question correct to get the overall probability of Jack passing the course based on his average score.
Based on these calculations, the correct answers to the questions are:
1. The probability that Jack gets 96% on the test is very low, and option A (0.000000001) is the closest value.
2. The probability that Jack will pass the course is also low, and option D (0.1814) is the closest value.