Question

rows represent the ratings students gave to the food.
columns represent the different food entrees at the cafeteria.

event a: choosing the meat entree
event b: choosing the vegan entree

probability of choosing the vegan entree:

p(b) =

probability of choosing the meat entree and choosing the vegan entree:

p(a and b) =

probability of choosing the meat entree or choosing the vegan entree:

p(a or b) =

Explanation:

Step1: Calculate \( P(B) \)

Probability formula: \( P(B)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).
Number of vegan entrees \( n(B) = 25 \), total number of entrees \( n(T)=60 \).
\( P(B)=\frac{25}{60}=\frac{5}{12}\)

Step2: Calculate \( P(A\cap B) \)

Since a student can't choose both meat and vegan entrees at the same - time (mutually exclusive events), \( n(A\cap B) = 0 \).
By the probability formula \( P(A\cap B)=\frac{n(A\cap B)}{n(T)} \), so \( P(A\cap B)=\frac{0}{60}=0\)

Step3: Calculate \( P(A\cup B) \)

Use the formula \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
Number of meat entrees \( n(A) = 12 \), so \( P(A)=\frac{12}{60}=\frac{1}{5}\), \( P(B)=\frac{25}{60}=\frac{5}{12}\), \( P(A\cap B) = 0 \).
\( P(A\cup B)=\frac{12 + 25}{60}=\frac{37}{60}\)

Answer:

\( P(B)=\frac{5}{12}\), \( P(A\cap B)=0\), \( P(A\cup B)=\frac{37}{60}\)