A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 70% of the questions correct

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To find the probability that the student passes the quiz with a grade of at least 70% correct, we need to determine the number of questions the student needs to answer correctly in order to pass.

First, let's determine how many questions the student needs to answer correctly to achieve a grade of at least 70%. Since there are ten questions in total, 70% of ten is 7. This means the student needs to answer at least seven questions correctly to pass the quiz.

Next, we need to find the probability of the student answering a certain number of questions correctly. Since each question is a true-false question, there are two possible answers: true or false. This means there is a 50% chance of guessing the correct answer for each question.

To find the probability of the student correctly answering a certain number of questions, we can use the binomial probability formula:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)

Where:
P(X=k) is the probability of getting k successes (in this case, k correct answers).
n is the total number of trials (in this case, ten questions).
k is the number of desired successes (in this case, at least seven correct answers).
p is the probability of success on a single trial (in this case, 0.5 since there's a 50% chance of guessing correctly).
nCk is the number of combinations of n items taken k at a time, which can be calculated using the formula n! / (k! * (n-k)!), where n! denotes the factorial of n.

In this case, we want to find the probability of the student answering at least seven questions correctly. This means we need to calculate the probabilities of getting exactly seven correct, exactly eight correct, nine correct, and ten correct answers, and then sum them up.

P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)

Let's calculate each of these probabilities using the formula mentioned above.

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