Question

math 120 exam 2 review chapters 5 - 7
a university conducted a survey of 390 undergraduate students regarding satisfaction with student government. results of the survey are shown in the table by class rank.

	freshman	sophomore	junior	senior	total
neutral	25	10	14	11	60
not satisfied	23	17	20	78	130
total	105	82	100	153	390

(a) (5.2) if a survey participant is selected at random, what is the probability that he or she is a junior?
(b) (5.2) if a survey participant is selected at random, what is the probability that he or she is satisfied or a junior?
(c) (5.4) if a junior is selected at random, what is the probability that he or she is satisfied?

1. a baseball team consists of three outfielders, four infielders, a pitcher, and a catcher.

(a) (5.5) in how many ways can a captain and assistant be chosen for this team?

Explanation:

Step1: Identify total number of participants

The total number of survey - participants is 390.

Step2: Calculate probability of selecting a junior (a)

The number of juniors is 276. The probability $P(\text{junior})$ is the number of juniors divided by the total number of participants. So $P(\text{junior})=\frac{276}{390}\approx0.7077$.

Step3: Calculate probability of being satisfied or a junior (b)

The number of satisfied participants is 242, the number of juniors is 276, and the number of satisfied juniors is 100. Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, we have $P(\text{satisfied}\cup\text{junior})=\frac{242 + 276-100}{390}=\frac{418}{390}\approx1.072$ (There is an error in the hand - written work in the picture for this part. The correct calculation: $n(\text{satisfied}\cup\text{junior})=n(\text{satisfied})+n(\text{junior})-n(\text{satisfied and junior})=242 + 276-100 = 418$, and $P=\frac{418}{390}$).

Step4: Calculate probability of being satisfied given junior (c)

The number of satisfied juniors is 66 and the number of juniors is 100. The conditional probability $P(\text{satisfied}|\text{junior})=\frac{n(\text{satisfied and junior})}{n(\text{junior})}=\frac{66}{100} = 0.66$.

Step5: Solve baseball team problem

A baseball team has 9 players. To choose a captain and an assistant, for the captain, there are 9 choices. After choosing a captain, for the assistant, there are 8 choices. By the multiplication principle, the number of ways to choose a captain and an assistant is $9\times8=72$.

Answer:

(a) $\frac{276}{390}\approx0.7077$
(b) $\frac{418}{390}$
(c) $\frac{66}{100}=0.66$
(d) 72