On a typical scantron exam, there are 4 choices for each problem (a, b, c, or d). If you are given an exam with 6 problems and you guessed randomly, what is them probability of getting at least half the problems right?
so you want 2 right + 3 right + 4 right
prob(correct guess) = 1/4
prob(wrong guess) = 3/4
prob as stated
= C(4,2)(1/4)^2 (3/4)^2 + C(4,3) (1/4)^3 (3/4)^1 + C(4,4)(1/4)^4 (3/4)^0
= 6(1/16)(9/16) + 4(1/64)(3/4) + 1/256
= 67/256
To find the probability of getting at least half the problems right when guessing randomly on a scantron exam, we can consider the number of problems we need to answer correctly.
In this case, we have 6 problems on the exam, so we need to answer at least 3 problems correctly to get at least half of them right.
Let's break down the possible outcomes for each number of correct answers:
1. If you answer only 3 problems correctly, you can choose 3 correct answers out of 6 problems in C(6, 3) = 20 ways. The remaining 3 answers will be incorrect, and there are C(4, 3) = 4 ways to choose one incorrect answer for each problem. Therefore, there are 20 * 4 = 80 ways to answer exactly 3 problems correctly.
2. If you answer 4 problems correctly, you can choose 4 out of 6 correct answers in C(6, 4) = 15 ways. The other 2 answers will be incorrect, and there are C(4, 2) = 6 ways to choose two incorrect answers. Thus, there are 15 * 6 = 90 ways to answer exactly 4 problems correctly.
3. If you answer 5 problems correctly, you can choose 5 out of 6 correct answers in C(6, 5) = 6 ways. The remaining answer will be incorrect, and there are C(4, 1) = 4 ways to choose one incorrect answer. So, there are 6 * 4 = 24 ways to answer exactly 5 problems correctly.
4. If you answer all 6 problems correctly, there is only one possible way to do so.
The total number of possible outcomes is given by the number of choices for each question, raised to the power of the number of questions. In this case, there are 4 choices for each of the 6 problems, so there are 4^6 = 4096 possible outcomes.
To calculate the probability, we need to sum up the number of favorable outcomes (ways of answering at least half the problems correctly) and divide it by the total number of outcomes.
The number of favorable outcomes is the sum of the ways to answer 3, 4, 5, or all 6 problems correctly: 80 + 90 + 24 + 1 = 195.
Therefore, the probability of getting at least half the problems right when guessing randomly is 195/4096, which is approximately 0.0476 or 4.76%.