Question

note: please make sure to properly format your answers. all dollar figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma ($2,354.67). all percentage values in the answers need to include a percentage sign (%). for all items without specific rounding instructions, round your answers to two decimal places, show both decimal places (5.00). the following table shows music preferences found by a survey of the faculty at a local university. express your answers in fraction form.

	country music (c)	rock music (r)	oldies (o)	total
southern u.s. (s)	70	50	44	164
total	81	138	93	312

a. find the probability that a randomly - selected person from this group likes country music.
b. what is the probability that a randomly - selected person from this group likes rock music and is from the north?
c. find the probability that a randomly - selected person from this group likes oldies given that they are from the south.
d. find p(r) in decimal form. round to two decimal places.
e. find p(s) in decimal form. round to two decimal places.
f. find p(r|s) and explain if events r and s are independent or associated events.

Explanation:

Step1: Recall probability formula

The probability of an event $A$ is $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

Step2: Calculate probability of liking country music (a)

The number of people who like country music is $81$, and the total number of people is $312$. So $P(C)=\frac{81}{312}=\frac{27}{104}$.

Step3: Calculate probability of liking rock music and being from the North (b)

The number of people who like rock music and are from the North is $88$, and the total number of people is $312$. So $P(R\cap N)=\frac{88}{312}=\frac{11}{39}$.

Step4: Calculate conditional - probability of liking oldies given from the South (c)

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. The number of people from the South is $164$, and the number of people from the South who like oldies is $44$. So $P(O|S)=\frac{44}{164}=\frac{11}{41}$.

Step5: Calculate $P(R)$ in decimal form (d)

The number of people who like rock music is $138$, and the total number of people is $312$. So $P(R)=\frac{138}{312}\approx0.44$.

Step6: Calculate $P(S)$ in decimal form (e)

The number of people from the South is $164$, and the total number of people is $312$. So $P(S)=\frac{164}{312}\approx0.53$.

Step7: Calculate $P(R|S)$ (f)

The number of people from the South who like rock music is $50$, and the number of people from the South is $164$. So $P(R|S)=\frac{50}{164}=\frac{25}{82}\approx0.30$. Events $R$ and $S$ are associated because $P(R|S)
eq P(R)$.

Answer:

a. $\frac{27}{104}$
b. $\frac{11}{39}$
c. $\frac{11}{41}$
d. $0.44$
e. $0.53$
f. $\frac{25}{82}$; Associated