All the fourth-graders in a certain elementary school took a standardized test. A total of 85% of the students were found to be proficient in reading, 78% were found to be proficient in mathematics, and 65% were found to be proficient in both reading and mathematics. A student is chosen at random. a. What is the probability that the student is proficient in mathematics but not in reading? b. What is the probability that the student is proficient in reading but not in mathematics? c. What is the probability that the student is proficient in neither reading nor mathematics?

The best way to solve this problem is by drawing a Venn diagram. Draw a rectangle representing all the fourth-graders. Draw two overlapping circles inside the rectangle. Let one circle represent proficiency in reading. This circle is 85% of the total area (including the overlap). And let the other circle represent proficiency in math. This circle is 78% of the total area (including the overlap). The overlap is 65% of the total area.

a. Since the overlap is 65%, and 78% are proficient in math, then the percent of all students who are proficient in math but not reading is the difference:

78% − 65% = 13%

b. Since the overlap is 65%, and 85% are proficient in reading, then the percent of all students who are proficient in reading but not math is the difference:

85% − 65% = 20%

c. The percent of students not proficient in reading or math is 100% minus the percent proficient in only reading minus the percent proficient in only math minus the percent proficient in both.

100% − 20% − 13% − 65% = 2%

See attached illustration (not to scale).