A quiz has 12 questions. What is the probability, as a percentage, that the student will answer at least 7 correctly?

Question

How to find the answer by using the sum of entries in the nth row of pascal triangle

Answer 1

Though not stated, assume the test is a True/False test, where the the probability of either choice is 50%.

Let P(n) represent the probability of getting n correct answers. We need to find P(7) + P(8) + P(9) + ... + P(12).

Pascal's Triangle is as follows:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

Notice that the sum of each row is a power of 2.

To use Pascal's triangle, we first find

P(7) = 792 / (1 + 12 + 66 + 220 + ... + 12 + 1) = 792 / 2¹²

Similarly,

P(8) = 495 / 2¹²

P(9) = 220 / 2¹²

P(10) = 66 / 2¹²

P(11) = 12 / 2¹²

P(12) = 1 / 2¹²

Hence, the answer is the sum of these probabilities.