a student takes a true false test that has 13 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(fewer than 4)
To find the probability of getting fewer than 4 questions correct, we need to determine the probability of getting 0, 1, 2, or 3 questions correct and then sum those probabilities.
Since the student is guessing randomly on each question, the probability of guessing correctly on a single question is 1/2 (since there are two options: true or false).
Now, let's calculate the probability of getting 0, 1, 2, or 3 questions correct.
Getting 0 questions correct means the student guesses incorrectly on all 13 questions. The probability of guessing incorrectly on a single question is 1/2, so the probability of getting 0 questions correct is (1/2)^13.
Getting 1 question correct means the student guesses correctly on one question and incorrectly on the remaining 12 questions. The probability of guessing correctly on one question is 1/2, so the probability of getting 1 question correct is (1/2)^1 * (1/2)^12.
Similarly, the probability of getting 2 questions correct is (1/2)^2 * (1/2)^11, and the probability of getting 3 questions correct is (1/2)^3 * (1/2)^10.
To find P(fewer than 4), we need to sum up these probabilities:
P(fewer than 4) = P(0) + P(1) + P(2) + P(3)
= (1/2)^13 + (1/2)^1 * (1/2)^12 + (1/2)^2 * (1/2)^11 + (1/2)^3 * (1/2)^10
Now, we can simplify and calculate the value:
P(fewer than 4) = (1/2)^13 + (1/2)^13 + (1/2)^13 + (1/2)^13
= 4 * (1/2)^13
≈ 0.0001220703125
Therefore, the probability of getting fewer than 4 questions correct is approximately 0.0001220703125.
To find the probability of fewer than 4 questions answered correctly, we need to calculate the probability of the student answering 0, 1, 2, or 3 questions correctly.
The probability of answering a question correctly by guessing is 1/2 since there are only two possible options (true or false).
Let's calculate the individual probabilities first:
P(X = 0) = (1/2)^13 = 1/8192
P(X = 1) = 13 * (1/2)^13 = 13/8192
P(X = 2) = (13C2) * (1/2)^13 = 78/8192
P(X = 3) = (13C3) * (1/2)^13 = 286/8192
To find P(fewer than 4), we can sum up these individual probabilities:
P(fewer than 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
= 1/8192 + 13/8192 + 78/8192 + 286/8192
= 378/8192
= 9/196
Therefore, the probability of answering fewer than 4 questions correctly is 9/196.
cases:
one right
one right
two right
three right
find prob of each case and add them up
I will do one of them:
prob(exactly 3 of 13 right)
= C(13,3) (1/2)^3 (1/2)^10
= 286(1/2)^13
= 286/8192
= 143/4098