Darby guesses the answers to six questions on the math portion of her college entrance exam. Each question is a true or false question. What is the approximate probability that at least 4 of her answers are correct?
A. 0.23
B. 0.34
C. 0.66
D. 0.78
To find the approximate probability that at least 4 of Darby's answers are correct, we can use the binomial probability formula.
The binomial probability formula is given by P(X=k) = nCk * p^k * (1-p)^(n-k), where:
- P(X=k) represents the probability of getting exactly k successes
- n is the number of trials (in this case, the number of questions)
- k is the number of successes (in this case, at least 4 answers correct)
- p is the probability of success on any individual trial (in this case, the probability of getting a correct answer)
In this case, since each question has only two possible outcomes (true or false), the probability of getting a correct answer is 0.5, and the probability of getting an incorrect answer is 1-0.5=0.5.
Now let's calculate the probability of at least 4 correct answers. We need to calculate the probabilities for 4, 5, and 6 correct answers, and then sum them up.
P(4) = 6C4 * (0.5)^4 * (1-0.5)^(6-4) = 15 * 0.0625 * 0.25 = 0.09375
P(5) = 6C5 * (0.5)^5 * (1-0.5)^(6-5) = 6 * 0.03125 * 0.5 = 0.09375
P(6) = 6C6 * (0.5)^6 * (1-0.5)^(6-6) = 1 * 0.015625 * 1 = 0.015625
Now let's sum up these probabilities:
P(at least 4 correct) = P(4) + P(5) + P(6) = 0.09375 + 0.09375 + 0.015625 = 0.203125
The approximate probability that at least 4 of Darby's answers are correct is approximately 0.203125. None of the given answer choices matches this probability exactly, so we would need additional answer choices to accurately determine the answer.