A student takes a 10-question true-false and randomly guesses the answer to each question. Suppose a correct answer is worth 1 point, an incorrect answer is worth -.5 points. Find the probability the student gets 7 or more points.
To solve this problem, we need to calculate the probability of the student getting 7 or more points by randomly guessing the answers to the 10 true-false questions.
First, let's calculate the probability of getting a question correct or incorrect:
1. Probability of getting a question correct: Since there are only two options (true or false), the probability of guessing the correct answer is 0.5 (since it's random).
2. Probability of getting a question incorrect: Similarly, the probability of guessing the incorrect answer is also 0.5.
Now, let's determine the number of ways the student can get 7 or more points. There are three possibilities to consider:
1. Getting exactly 7 questions correct: This can be achieved by choosing 7 questions correctly out of the total 10 questions and getting the remaining 3 questions incorrect. The number of ways to do this can be calculated using the binomial coefficient formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of questions (10) and k is the number of questions answered correctly (7). Therefore, the number of ways to get exactly 7 correct answers is C(10, 7).
2. Getting exactly 8 questions correct: Similarly, by choosing 8 questions correctly out of the total 10 questions and getting the remaining 2 questions incorrect, we can calculate the number of ways to get exactly 8 correct answers using the binomial coefficient formula: C(10, 8).
3. Getting all 10 questions correct: Finally, we consider the possibility of getting all 10 questions correct.
To calculate the probability of each of these cases occurring, we need to multiply the number of ways of getting each case by the probabilities of getting each answer correct or incorrect:
1. Probability of getting exactly 7 questions correct: (C(10, 7) * (0.5)^7 * (0.5)^3) = C(10, 7) * (0.5)^10
2. Probability of getting exactly 8 questions correct: (C(10, 8) * (0.5)^8 * (0.5)^2) = C(10, 8) * (0.5)^10
3. Probability of getting all 10 questions correct: (0.5)^10
Finally, to find the probability of getting 7 or more points, we sum up the probabilities of these three cases:
Probability = Probability of getting exactly 7 questions correct + Probability of getting exactly 8 questions correct + Probability of getting all 10 questions correct
Probability = C(10, 7) * (0.5)^10 + C(10, 8) * (0.5)^10 + (0.5)^10.