Question

a taste test asks people from Texas and California which pasta they prefer, brand A or brand B. this table shows the results

Answer 1

Answer: Option D

Explanation:

First we assign names to events:

Event C: The selected people are from California

Event A: The selected person prefers the A mark

Now notice in the table that the total number of people is: 275.

Then, the number of people who prefer the A mark is: 176

The number of people who are from California is: 150

The number of people in California who prefer the A brand is: 96

Then we have that:

`P(C)=(150)/(275)=(6)/(11)=0.55`

`P(A)=(176)/(275)=(16)/(25)`

`P (C\ and\ A) = (96)/(275)=0.3491`

Then:

`P(C|A)=(P(C\ and\ A))/(P(A))\\\\P(C|A)=((96)/(275))/((16)/(25))=(6)/(11)=0.55`

By definition two events C and A are independent if and only if:

`P (C\ and\ A) = P (A)*P (C)`

Then, if A and C are independent events, it must be fulfilled that:

`P(C|A)=(P(A)*P(C))/(P(A))\\\\P(C|A)=P(C)`

Note that
`P (C) = 0.55` and
`P (C | A) = 0.55`

So:

`P (C | A) = P (C)`

Therefore the events are independent and
`P (C) =P (C | A) = 0.55`

Answer 2

D.

Explanation:

Two events A and B are independent when

`P(A|B)=P(A),`

where
`P(A|B)` is the probability that event A occurs given that event B has occured.

A = randomly selected person is from California

B = randomly selected person preferred brand A

A|B = being a person from California and preferring brand A.

Hence,

`P(A)=(150)/(275)=(6)/(11)\approx 0.55\\ \\P(B)=(176)/(275)=(16)/(25)=0.64\\ \\P(A|B)=(96)/(176)=(6)/(11)\approx 0.55`

Since
`P(A)=P(A|B),` events are independent.