Question

question 10 of 10
a taste test asks people from texas and california which pasta they prefer, brand a or brand b. this table shows the results.
a person is randomly selected from those tested.
are being from california and preferring brand a independent events? why or why not?
a. yes, they are independent because ( p(\text{california}) approx 0.55 ) and ( p(\text{california|brand a}) = 0.64 ).
b. no, they are not independent because ( p(\text{california}) approx 0.55 ) and ( p(\text{california|brand a}) approx 0.64 ).
c. yes, they are independent because ( p(\text{california}) approx 0.55 ) and ( p(\text{california|brand a}) approx 0.55 ).

Explanation:

Step1: Calculate \(P(\text{California})\)

The formula for probability is \(P=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\).
The number of people from California is \(150\), and the total number of people tested is \(275\).
So, \(P(\text{California})=\frac{150}{275}\approx0.55\)

Step2: Calculate \(P(\text{California}|\text{brand A})\)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).
Here, \(n(\text{California}\cap\text{brand A}) = 96\), \(n(\text{brand A})=176\)
So, \(P(\text{California}|\text{brand A})=\frac{96}{176}\approx0.55\)

Step3: Check for independence

Two events \(A\) and \(B\) are independent if \(P(A|B)=P(A)\)
Since \(P(\text{California})\approx0.55\) and \(P(\text{California}|\text{brand A})\approx0.55\), the events are independent.

Answer:

C. Yes, they are independent because \(P(\text{California})\approx0.55\) and \(P(\text{California}|\text{brand A})\approx0.55\)