Part A of your history test has 15 multiple choice questions. Each question has four choices. Part B has 10 true/false questions. How many ways are there to answer the 15 multiple choice questions? How many ways are there to answer the 10 true or questions? How many are there to answer all 25 questions? If you guess the answer to each question, what is the probability that you will get them all right?
15 MULTIPLE CHOICE : 4^15 = 1073741824
10 T/F CHOICE : 2^10 = 1024
25 QUESTIONS : (4^15)(2^10)
1.363e-10
To find the number of ways to answer the multiple-choice questions, we need to calculate 4 raised to the power of 15, since each question has four choices and there are 15 questions.
Number of ways to answer multiple-choice questions = 4^15 = 1,073,741,824 ways.
To find the number of ways to answer the true/false questions, we need to calculate 2 raised to the power of 10, since each question has two choices (true or false) and there are 10 questions.
Number of ways to answer true/false questions = 2^10 = 1,024 ways.
To find the number of ways to answer all 25 questions, we need to multiply the number of ways to answer multiple-choice questions with the number of ways to answer true/false questions.
Number of ways to answer all questions = 1,073,741,824 * 1,024 = 1,099,511,627,776 ways.
If you guess the answer to each question, there is a 1 in 4 chance of getting the multiple-choice questions correctly and a 1 in 2 chance of getting the true/false questions correctly.
Therefore, the probability of getting all the questions correct is (1/4)^15 * (1/2)^10.
Probability of getting all questions correct = (1/1,073,741,824) * (1/1,048,576) = 1 in 1,123,549,439,964,491,712,256 or approximately 8.903 * 10^-23.
To find the number of ways to answer the multiple-choice and true/false questions, we can use the concept of permutations.
For the multiple-choice questions:
Each question has 4 choices, so for each question, there are 4 possible ways to choose an answer.
Since there are 15 multiple-choice questions, the total number of ways to answer them all is 4^15. (4 raised to the power of 15)
For the true/false questions:
Each question has 2 choices, so for each question, there are 2 possible ways to choose an answer.
Since there are 10 true/false questions, the total number of ways to answer them all is 2^10. (2 raised to the power of 10)
To find the number of ways to answer all 25 questions, we multiply the total number of ways to answer each type of question:
Total number of ways to answer all 25 questions = (4^15) * (2^10) = 1,073,741,824 * 1,024 = 1,099,511,627,776.
If you guess the answer to each question, the probability of getting them all right is the ratio of the number of ways to answer all questions correctly (1 way) to the total number of possible ways to answer (1,099,511,627,776 ways):
Probability of getting all questions right = 1 / 1,099,511,627,776 ≈ 9.09 × 10^(-13).