To get an A grade on the test, you need a total score of more than 16 points. One of the students knows the correct answer to 6 of the 20 questions. The rest she guesses at random by tossing a coin (one toss per question, as in 4B). What is the chance that she gets an A grade on the test?

Question

To get an A grade on the test, you need a total score of more than 16 points. One of the students knows the correct answer to 6 of the 20 questions. The rest she guesses at random by tossing a coin (one toss per question, as in 4B). What is the chance that she gets an A grade on the test?

Answer 1

To find the chance that the student gets an A grade on the test, we need to calculate the probability of her scoring more than 16 points.

Let's break down the problem step by step:

1. The student knows the correct answer to 6 out of the 20 questions. This means she is guaranteed to score 6 points.

2. For the remaining 14 questions, she will guess at random by tossing a coin. Since each question has two possible answers (either she gets it right or wrong), the probability of her guessing the correct answer for each question is 1/2.

3. To calculate the probability of scoring more than 16 points, we need to consider all the possible combinations of correct and incorrect guesses for the remaining 14 questions.

4. The number of ways to choose the correct answers out of 14 questions is a combination problem. We can calculate it using the formula C(n, r) = n! / (r!(n-r)!), where n is the total number of questions and r is the number of correct answers.

Applying the formula, we get C(14, x), where x is the number of correct guesses. Since we want to calculate the probability of scoring more than 16 points, we sum up all the combinations from x=11 to x=14.

5. For each combination, we need to calculate the probability of getting those specific number of correct guesses. Since the probability of guessing correctly for each question is 1/2, the probability of getting x correct guesses out of 14 questions is (1/2)^x.

6. We sum up the probabilities for all the combinations from x=11 to x=14 to find the total probability of scoring more than 16 points.

Now, let's calculate the probability:

P(score > 16) = P(11 correct) + P(12 correct) + P(13 correct) + P(14 correct)

P(11 correct) = C(14, 11) * (1/2)^11
P(12 correct) = C(14, 12) * (1/2)^12
P(13 correct) = C(14, 13) * (1/2)^13
P(14 correct) = C(14, 14) * (1/2)^14

We can plug the values into the formula and calculate the probability.