Question

Exercise 4.6. [Purpose: Recognize and work with the fact that Equation 4.9 can be solved for the joint probability, which wil be crucial for developing Bayes' theorem.] School children were surveyed regarding their favorite foods. Of the total sample, 20% were 1st graders, 20% were 6th graders, and 60% were 11th graders For each grade, the following table shows the proportion of respondents that chose each of three foods as their favorite: lce cream Fruit French fries 1st graders0.3 0.6 0.1 0.3 0.1 6th graders 0.6 11th graders |0.3 0.10.6 From that information, construct a table of joint probabilities of grade and favorite food. Also, say whether grade and favorite food are independent or not, and how you ascertained the answer. Hint: You are given p(grade) and p(foodlgrade). You need to determine p(grade,food)

Answer 1

It is given that

P(1st grader) = 0.2. P(6th grader) = 0.2, P(11th grader) = 0.6

We have to find

P (grade, food)- P( food grade) P (grade)

For example

P (1st grader, Ice cream)P( Ice cream | 1st grader)P(1st grader) P(1st grader , Ice cream) = (O.3)(0.2) 0.06

In this way other joint probabilities could be calculated.

The joint PMF table is

Grade/Food	Ice cream	Fruit	French Fries
1st grader	0.06	0.12	0.02	0.2
6th grader	0.12	0.06	0.02	0.2
11th grader	0.18	0.06	0.36	0.6
	0.36	0.24	0.40	1.0

We can see that

P (grade, food) 1

As test for independence

P (1st grader, Ice cream) 0.06 P (1st grader )P( Ice cream) - 0.2 x 0.36

So the food and grades are not independent.