Question

in a class of 60 students, 30 are democrats, 14 are business majors, and 8 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

There are 30 Democrats and 8 of them are business majors. So number of Democrats who are not business majors is $30 - 8=22$.

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

Total number of students is 60. Probability $P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. So $P(\text{a Democrat who is not a business major})=\frac{22}{60}=\frac{11}{30}$.

Step3: Find number of students who are either Democrat or business major

Use the formula $n(A\cup B)=n(A)+n(B)-n(A\cap B)$. Here $n(A)$ (number of Democrats) = 30, $n(B)$ (number of business - majors) = 14 and $n(A\cap B)$ (number of Democrats who are business majors) = 8. So $n(A\cup B)=30 + 14-8=36$.

Step4: Find number of students who are neither Democrat nor business major

Total number of students is 60. So number of students who are neither Democrat nor business major is $60 - 36 = 24$.

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

Probability $P=\frac{24}{60}=\frac{2}{5}$.

Answer:

a. $\frac{11}{30}$
b. $\frac{2}{5}$