On a true-false test, each question has exactly one correct answer: true, or false. A student knows the correct answer to 70% of the questions on the test. Each of the remaining answers she guesses at random, independently of all other answers. After the test has been graded, one of the questions is picked at random. Given that she got the answer right, what is the chance that she knew the answer?
unanswered
0.824
its not correct
you need 4 numbers after the ,
0.8235
To find the chance that the student knew the answer given that she got it right, we can use Bayes' Theorem. Bayes' Theorem states that the probability of an event A given that event B has occurred is equal to the probability of event B given that event A has occurred, multiplied by the probability of event A, divided by the probability of event B.
In this case, event A is the student knowing the correct answer, and event B is the student getting the answer right. We want to find the probability that event A occurred (the student knowing the answer) given that event B occurred (the student got the answer right).
Let's assign some variables:
P(A) = probability that the student knew the answer = 0.7 (since the student knows the correct answer to 70% of the questions)
P(~A) = probability that the student did not know the answer = 0.3 (since the student guesses randomly on 30% of the questions)
P(B|A) = probability of getting the answer right given that the student knew the answer = 1 (since if the student knows the answer, she will get it right)
P(B|~A) = probability of getting the answer right given that the student did not know the answer = 0.5 (since the student has a 50% chance of randomly guessing the answer right)
We want to find P(A|B), the probability that the student knew the answer given that she got it right.
Using Bayes' Theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
Now let's calculate P(B), the probability of getting the answer right:
P(B) = (P(B|A) * P(A)) + (P(B|~A) * P(~A))
= (1 * 0.7) + (0.5 * 0.3)
= 0.7 + 0.15
= 0.85
Now we can substitute the values into Bayes' Theorem:
P(A|B) = (1 * 0.7) / 0.85
= 0.7 / 0.85
≈ 0.8235
Therefore, given that the student got the answer right, there is about an 82.35% chance that she knew the answer.