a test consists of true/false questions .to pass the test a student must answer at least 6questions correctly .if a student guesses on each question,what is the probability that the student will pass the test

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To calculate the probability that the student will pass the test by guessing on each question, we need to determine the probability of answering at least 6 questions correctly out of the total number of questions.

Since each question has only two possible answers (true/false), the probability of guessing the correct answer for each question is 1/2.

Let's use binomial probability to find the probability of passing the test. The binomial probability formula is:
P(x) = C(n, x) * p^x * q^(n-x)

Where:
P(x) is the probability of getting exactly x successes
C(n, x) is the number of combinations of n items taken x at a time
p is the probability of success on a single trial
q is the probability of failure on a single trial
n is the total number of trials or questions

In this case, we want to find the probability of answering at least 6 questions correctly, so we need to calculate the sum of the probabilities for 6, 7, 8, 9, and 10 correct answers out of 10 total questions.

P(pass) = P(6) + P(7) + P(8) + P(9) + P(10)

Using the binomial probability formula, with n = 10, p = 1/2, and q = 1/2:

P(pass) = C(10, 6) * (1/2)^6 * (1/2)^(10-6) + C(10, 7) * (1/2)^7 * (1/2)^(10-7) + C(10, 8) * (1/2)^8 * (1/2)^(10-8) + C(10, 9) * (1/2)^9 * (1/2)^(10-9) + C(10, 10) * (1/2)^10 * (1/2)^(10-10)

Calculating each term:

P(6) = C(10, 6) * (1/2)^6 * (1/2)^(10-6) = 210 * (1/2)^6 * (1/2)^4
P(7) = C(10, 7) * (1/2)^7 * (1/2)^(10-7) = 120 * (1/2)^7 * (1/2)^3
P(8) = C(10, 8) * (1/2)^8 * (1/2)^(10-8) = 45 * (1/2)^8 * (1/2)^2
P(9) = C(10, 9) * (1/2)^9 * (1/2)^(10-9) = 10 * (1/2)^9 * (1/2)^1
P(10) = C(10, 10) * (1/2)^10 * (1/2)^(10-10) = 1 * (1/2)^10 * (1/2)^0

Finally, calculating the sum:

P(pass) = P(6) + P(7) + P(8) + P(9) + P(10)

You can evaluate this expression to find the probability that the student will pass the test by guessing.

The probability of answering a true-false question correctly by just guessing is 1/2 or .5. The probability of getting all the questions correct (in this case, 6) is found by multiplying the probabilities of the individual events.

I'll let you do the calculations. I hope this helps.

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