Question

question 6 of 8 select the correct answer. a juice tasting was conducted for three randomly - selected groups of people. the results of the data are shown in the table. using the data, conclusions are made based on calculated probabilities of a randomly - selected person from the survey.

	group 1	group 2	group 3	total
juice b	28	59	47	134
juice c	40	44	49	133
total	100	120	130	350

given the person is from any group, they select juice a. given the person is from any group, they select juice c. given the person selects juice b, the probabilities of the person being from any group are similar. given the person selects juice c, the probabilities of the person being from any group are similar.

Explanation:

Step1: Recall probability formula

The probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In terms of frequency - based probability from a contingency table, if we want to find the probability that a person is from a certain group given they selected a certain juice, we use the formula $P(\text{Group}|\text{Juice})=\frac{\text{Number of people in group and selected juice}}{\text{Number of people who selected the juice}}$.

Step2: Calculate probability of selecting juice A

The total number of people who selected juice A is $83$. The probability that a person selected from any group is juice A is $P(\text{Juice A})=\frac{83}{350}$.

Step3: Calculate probability of selecting juice B

The total number of people who selected juice B is $134$. The probability that a person selected from any group is juice B is $P(\text{Juice B})=\frac{134}{350}$.

Step4: Calculate probability of selecting juice C

The total number of people who selected juice C is $133$. The probability that a person selected from any group is juice C is $P(\text{Juice C})=\frac{133}{350}$.

Step5: Calculate conditional probabilities for each juice - group combination

For example, if we want to find the probability that a person is from Group 1 given they selected juice A, $P(\text{Group 1}|\text{Juice A})=\frac{32}{83}$. Similarly, we can calculate other conditional probabilities for all juice - group combinations. But we are interested in the overall probabilities of selecting a juice and being from a group.
Let's assume we want to check the statement about the probabilities of a person being from any group given they selected a particular juice.
For juice A:
$P(\text{Group 1}|\text{Juice A})=\frac{32}{83}\approx0.386$, $P(\text{Group 2}|\text{Juice A})=\frac{17}{83}\approx0.205$, $P(\text{Group 3}|\text{Juice A})=\frac{34}{83}\approx0.410$
For juice B:
$P(\text{Group 1}|\text{Juice B})=\frac{28}{134}\approx0.209$, $P(\text{Group 2}|\text{Juice B})=\frac{59}{134}\approx0.440$, $P(\text{Group 3}|\text{Juice B})=\frac{47}{134}\approx0.351$
For juice C:
$P(\text{Group 1}|\text{Juice C})=\frac{40}{133}\approx0.301$, $P(\text{Group 2}|\text{Juice C})=\frac{44}{133}\approx0.331$, $P(\text{Group 3}|\text{Juice C})=\frac{49}{133}\approx0.368$

We can see that given the person selects juice C, the probabilities of the person being from any group are:
$P(\text{Group 1}|\text{Juice C})=\frac{40}{133}$, $P(\text{Group 2}|\text{Juice C})=\frac{44}{133}$, $P(\text{Group 3}|\text{Juice C})=\frac{49}{133}$ which are relatively close in value compared to other juice - group combinations.

Answer:

Given the person selects juice C, the probabilities of the person being from any group are similar.