There are six true-false questions on a quiz. Not knowing any of the answers, a student guesses randomly. Find the probability of each event. Exactly two answers are correct. Exactly six answers are correct. At least five are correct. At least three answers are correct.
To find the probability of each event, we need to understand the number of outcomes for each event and the total number of possible outcomes.
Given that there are six true-false questions, each question has two possible answers (true or false). Thus, there are 2^6 = 64 possible outcomes for the six questions.
Now, let's calculate the probability for each event:
1. Exactly two answers are correct:
To find this probability, we need to choose 2 questions out of 6 in which the student answers correctly, and the remaining 4 questions will be answered incorrectly. The number of ways to choose 2 questions out of 6 is denoted by 6C2 = 15. Additionally, for each of these 15 combinations, the student answers correctly on the chosen questions and incorrectly on the remaining, resulting in a probability of (1/2)^2 * (1/2)^4 = 1/64. Therefore, the probability of exactly two answers being correct is 15 * 1/64 = 15/64.
2. Exactly six answers are correct:
Since the student is randomly guessing, the probability of getting all six answers correct is (1/2)^6 = 1/64.
3. At least five answers are correct:
To find this probability, we sum up the probabilities of getting five or six answers correct. We have already determined that the probability of getting exactly six correct is 1/64. Now, to find the probability of getting exactly five answers correct, we need to choose 5 questions out of 6 to answer correctly, resulting in a probability of 6C5 * (1/2)^5 * (1/2)^1 = 6/64 = 3/32. Adding the probabilities, we get 1/64 + 3/32 = 5/64. Therefore, the probability of at least five answers being correct is 5/64.
4. At least three answers are correct:
To find this probability, we sum up the probabilities of getting three, four, five, or six answers correct. The probability of getting exactly three correct answers is 6C3 * (1/2)^3 * (1/2)^3 = 20/64. The probability of getting exactly four correct answers is 6C4 * (1/2)^4 * (1/2)^2 = 15/64. We have already determined the probability of at least five correct answers (5/64). Adding these probabilities, we get 20/64 + 15/64 + 5/64 = 40/64 = 5/8. Therefore, the probability of at least three answers being correct is 5/8.
In summary:
- The probability of exactly two answers being correct is 15/64.
- The probability of exactly six answers being correct is 1/64.
- The probability of at least five answers being correct is 5/64.
- The probability of at least three answers being correct is 5/8.