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
P(at least 4 correct)
= P(4 correct) + P(5 correct) + P(6 correct)
= 6C4 (½)^4 (½)^2 + 6C5 (½)^5 (½)^1 + 6C6 (½)^6 (½)^0
... four correct answers, two incorrect, in any order plus five correct answers, one incorrect, in any order, plus six correct answers, zero incorrect, and only one possible order ...
= 15 (½)^6 + 6 (½)^6 + 1 (½)^6
= (15+6+1) / (2^6)
= 22 / 64
= 11 / 32
= 0.34375
How did you get this?
= 6C4 (½)^4 (½)^2 + 6C5 (½)^5 (½)^1 + 6C6 (½)^6 (½)^0
To find the approximate probability that at least 4 of Darby's answers are correct, we can use the binomial probability formula.
The formula for calculating the probability of exactly k successes in n independent Bernoulli trials (where each trial has a constant probability of success, p) is:
P(k) = (nCk) * p^k * (1 - p)^(n - k)
In this case, we want to find the probability of at least 4 correct answers, which means either 4, 5, or 6 correct answers. So we need to calculate the probability of 4, 5, and 6 correct answers separately, and then add them together.
Let's assume that Darby has a 50% chance of guessing the correct answer for each question, since the questions are multiple-choice and she has only two options (true or false).
Calculating the probability of 4 correct answers:
P(4) = (6C4) * (0.5)^4 * (0.5)^(6 - 4) = 15 * 0.0625 * 0.0625 = 0.0586
Calculating the probability of 5 correct answers:
P(5) = (6C5) * (0.5)^5 * (0.5)^(6 - 5) = 6 * 0.03125 * 0.5 = 0.09375
Calculating the probability of 6 correct answers:
P(6) = (6C6) * (0.5)^6 * (0.5)^(6 - 6) = 1 * 0.015625 * 1 = 0.015625
Now, we can add up the probabilities of 4, 5, and 6 correct answers to find the probability of at least 4 correct answers:
P(at least 4) = P(4) + P(5) + P(6) = 0.0586 + 0.09375 + 0.015625 = 0.167975
Since we're looking for an approximate probability, we can round the value to two decimal places. Therefore, the approximate probability that at least 4 of Darby's answers are correct is approximately 0.17.
None of the given answer choices match this result. Therefore, none of the options A, B, C, or D are correct.