A study of 180 college students categorized them by their major field of study (STEM or Hu… — SAT

A study of $180$ college students categorized them by their major field of study (STEM or Humanities) and whether they participate in undergraduate research. The table below shows the partial results of the study, where $a$ , $b$ , $c$ , and $d$ represent the number of students in each category.

Major	Participates in Research	Does Not Participate in Research	Total
STEM	$a$	$b$	$110$
Humanities	$c$	$d$	$70$
Total	$80$	$100$	$180$

Given that a student selected at random is a Humanities major, the probability that the student participates in undergraduate research is $\frac{3}{7}$ . If a student who participates in undergraduate research is selected at random, what is the probability that the student is a STEM major?

$\frac{5}{8}$ Answer
B
$\frac{5}{18}$
C
$\frac{5}{11}$
D
$\frac{1}{2}$

Answer

The correct probability is

\frac{5}{8}

To find the probability that a student is a STEM major given they participate in undergraduate research, we first determine the missing values in the table. The total number of Humanities majors is

70

. Since the probability that a Humanities major participates in research is

\frac{3}{7}

, the number of Humanities majors in research is

c = \frac{3}{7} \times 70 = 30

. Using the research column total, the number of STEM majors in research is

a = 80 - c = 80 - 30 = 50

. The probability that a student is a STEM major given that they participate in research is the ratio of STEM research participants to total research participants:

\frac{a}{a+c} = \frac{50}{80} = \frac{5}{8}

Step-by-Step Solution

Find the number of Humanities majors who participate in research (

c

c = 30

The table shows there are

70

Humanities majors in total. We are given that the probability of a Humanities major participating in research is

\frac{3}{7}

. Therefore,

\frac{c}{70} = \frac{3}{7}

, which simplifies to

c = 30

Find the number of STEM majors who participate in research (

a

a = 50

The total number of students who participate in research is

80

. Since

a + c = 80

and

c = 30

, we have

a + 30 = 80

, which means

a = 50

Calculate the conditional probability that a student who participates in research is a STEM major.

\frac{5}{8}

We want to find the probability of selecting a STEM major given that the student participates in research. This is the ratio of STEM majors who participate in research (

a = 50

) to the total number of students who participate in research (

80

). The probability is

\frac{50}{80} = \frac{5}{8}

Key Concept

Conditional Probability in Two-Way Tables

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