Question

in a class of 46 students, 22 are democrats, 18 are business majors, and 3 of the business majors are democrats. if one student is randomly selected from the class, find the probability of choosing:
a. a democrat who is not a business major
b. a student who is neither a democrat nor a business major
a. p(a democrat who is not a business major)=
(type an integer or a simplified fraction.)
b. p(a student who is neither a democrat nor a business major)=
(type an integer or a simplified fraction.)

Explanation:

Step1: Find number of Democrats who are not business majors

We know there are 22 Democrats and 3 of them are business majors. So the number of Democrats who are not business majors is $22 - 3=19$.

Step2: Calculate probability of choosing a Democrat who is not a business major

The total number of students is 46. The probability $P$ is the number of favorable outcomes divided by the total number of outcomes. So $P(\text{a Democrat who is not a business major})=\frac{19}{46}$.

Step3: Find number of students who are either Democrats or business majors

We use the formula $n(A\cup B)=n(A)+n(B)-n(A\cap B)$. Here, $n(A) = 22$ (Democrats), $n(B)=18$ (business - majors) and $n(A\cap B)=3$. So $n(A\cup B)=22 + 18-3=37$.

Step4: Calculate number of students who are neither Democrats nor business majors

The total number of students is 46. So the number of students who are neither Democrats nor business majors is $46 - 37 = 9$.

Step5: Calculate probability of choosing a student who is neither a Democrat nor a business major

The probability $P$ is the number of favorable outcomes divided by the total number of outcomes. So $P(\text{a student who is neither a Democrat nor a business major})=\frac{9}{46}$.

Answer:

a. $\frac{19}{46}$
b. $\frac{9}{46}$