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 (5.06). 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

Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Step2: Calculate P(C) for part 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 P(R and N) for part 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 P(O|S) for part c

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

Step5: Calculate P(R) for part 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) for part 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) for part f

$P(R|S)=\frac{P(R\cap S)}{P(S)}$. The number of people who like rock music and are from the South is 50, and the number of people from the South is 164. So $P(R|S)=\frac{50}{164}=\frac{25}{82}\approx0.30$. If $R$ and $S$ are independent, then $P(R|S) = P(R)$. Since $P(R)\approx0.44$ and $P(R|S)\approx0.30$, $P(R|S)
eq P(R)$, so events $R$ and $S$ are associated events.

Answer:

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