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:

D.

Step-by-step 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,

$P(A)=\dfrac{150}{275}=\dfrac{6}{11}\approx 0.55\\ \\P(B)=\dfrac{176}{275}=\dfrac{16}{25}=0.64\\ \\P(A|B)=\dfrac{96}{176}=\dfrac{6}{11}\approx 0.55$

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

Answer 2

Answer: Option D

Step-by-step 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:

$P(C)=\frac{150}{275}=\frac{6}{11}=0.55$

$P(A)=\frac{176}{275}=\frac{16}{25}$

$P (C\ and\ A) = \frac{96}{275}=0.3491$

Then:

$P(C|A)=\frac{P(C\ and\ A)}{P(A)}\\\\P(C|A)=\frac{\frac{96}{275}}{\frac{16}{25}}=\frac{6}{11}=0.55$

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:

$P(C|A)=\frac{P(A)*P(C)}{P(A)}\\\\P(C|A)=P(C)$

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

So:

$P (C | A) = P (C)$

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