2015 AMC 12B Problem 15

Problem 15 of 25IntermediateCounting & Probability

At Rachelle’s school an A counts

4

points, a B

3

points, a C

2

points, and a D

1

point. Her GPA on the four classes she is taking is computed as the total sum of points divided by

4.

She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a

\dfrac16

chance of getting an A in English, and a

\dfrac14

chance of getting a B. In History, she has a

\dfrac14

chance of getting an A, and a

\dfrac13

chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least

3.5?

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Solution

Math and Science give

8

points, so Rachelle needs at least

6

more from English and History. The chance of a C is

1 - \dfrac16 - \dfrac14 = \dfrac{7}{12}

in English and

1 - \dfrac14 - \dfrac13 = \dfrac{5}{12}

in History. Working over a denominator of

144:

8

points has probability

\dfrac16\cdot\dfrac14 = \dfrac{6}{144};

7

points has

\dfrac16\cdot\dfrac13 + \dfrac14\cdot\dfrac14 = \dfrac{17}{144};

and

6

points has

\dfrac16\cdot\dfrac{5}{12} + \dfrac14\cdot\dfrac13

+ \dfrac{7}{12}\cdot\dfrac14 = \dfrac{43}{144}.

The total is

\dfrac{6 + 17 + 43}{144} = \dfrac{66}{144} = \dfrac{11}{24}.

Thus, the correct answer is D.