Question

1. middle school values researchers carried out a survey of fourth-, fifth-, and sixth - grade students in michigan. students were asked whether good grades, athletic ability, or being popular was most important to them. this two - way table summarizes the survey data.

most important\grade|4th grade|5th grade|6th grade|total
--|--|--|--|--
grades|49|50|69|168
athletic|24|36|38|98
popular|19|22|28|69
total|92|108|135|335
suppose we select one of these students at random. whats the probability that:
a. the student is a sixth grader or a student who rated good grades as important?
b. the student is not a sixth grader and did not rate good grades as important?

1. disks of four colors a jar contains 36 disks: 9 each of four colors - red, green, blue, and yellow. each set of disks of the same color is numbered from 1 to 9. suppose you draw one disk at random from the jar. define events b: get a blue disk, and e: get a disk with the number 8.

a. make a two - way table that describes the sample space in terms of events b and e.
b. find p(b) and p(e).
c. write the event “blue eight” in symbolic form. then find the probability of this event.

Explanation:

Step1: Calculate total number of students

The total number of students is 335 (from the table total).

Step2: Calculate number of sixth - graders or students who rated good grades as important

Let \(A\) be the set of sixth - graders and \(B\) be the set of students who rated good grades as important.
\(n(A)=135\), \(n(B) = 168\), \(n(A\cap B)=69\).
Using the formula \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\), we have \(n(A\cup B)=135 + 168-69=234\).
The probability \(P(A\cup B)=\frac{n(A\cup B)}{n(\text{total})}=\frac{234}{335}\approx0.7\)

Step3: Calculate number of students who are not sixth - graders and did not rate good grades as important

The number of students who are not sixth - graders is \(335 - 135 = 200\).
The number of students who did not rate good grades as important is \(335-168 = 167\).
The number of students who are sixth - graders and rated good grades as important is 69.
The number of students who are not sixth - graders and did not rate good grades as important is \(335-(135 + 168 - 69)=101\).
The probability is \(\frac{101}{335}\approx0.3\)

Step4: Make two - way table for disk problem

	\(E\) (Disk with number 8)	\(\overline{E}\) (Disk not with number 8)	Total
\(\overline{B}\) (Not blue disk)	3	24	27
Total	4	32	36

Step5: Calculate \(P(B)\) and \(P(E)\)

The probability \(P(B)=\frac{n(B)}{n(\text{total})}=\frac{9}{36}=\frac{1}{4}\)
The probability \(P(E)=\frac{n(E)}{n(\text{total})}=\frac{4}{36}=\frac{1}{9}\)

Step6: Write "blue eight" in symbolic form and find its probability

The event "blue eight" is \(B\cap E\).
The probability \(P(B\cap E)=\frac{n(B\cap E)}{n(\text{total})}=\frac{1}{36}\)

Answer:

a. \(\frac{234}{335}\approx0.7\)
b. \(\frac{101}{335}\approx0.3\)
c. Symbolic form: \(B\cap E\), Probability: \(\frac{1}{36}\)
d. Two - way table:

	\(E\) (Disk with number 8)	\(\overline{E}\) (Disk not with number 8)	Total
\(\overline{B}\) (Not blue disk)	3	24	27
Total	4	32	36

e. \(P(B)=\frac{1}{4}\), \(P(E)=\frac{1}{9}\)