73.) 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 |
| --- | --- | --- | --- | --- |
| northern u.s. (n) | 11 | 88 | 49 | 148 |
| 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.
g. zoe and lisa are having a disagreement about conditional probability. zoe thinks that ( p(a|b) = p(b|a) ) for any two events ( a ) and ( b ). lisa says that ( p(a|b) ) and ( p(b|a) ) do not have to be equal. pick two events from the table in problem 5 and determine who is correct.

Question

73.) 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 |
| --- | --- | --- | --- | --- |
| northern u.s. (n) | 11 | 88 | 49 | 148 |
| 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.
g. zoe and lisa are having a disagreement about conditional probability. zoe thinks that ( p(a|b) = p(b|a) ) for any two events ( a ) and ( b ). lisa says that ( p(a|b) ) and ( p(b|a) ) do not have to be equal. pick two events from the table in problem 5 and determine who is correct.

Accepted Answer

### Part a
# Explanation:
## Step1: Recall probability formula
The probability of an event $ E $ is $ P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} $. For liking country music, the number of favorable outcomes is the total number of people who like country music, and the total number of outcomes is the total number of people surveyed.
## Step2: Identify values from table
From the table, the total number of people who like country music (Country Music Total) is $ 81 $, and the total number of people surveyed is $ 312 $.
## Step3: Calculate probability
So, $ P(\text{Country Music})=\frac{81}{312} $. Simplify the fraction: divide numerator and denominator by 3, we get $ \frac{27}{104} $.

# Answer: $\frac{27}{104}$

### Part b
# Explanation:
## Step1: Recall probability formula for intersection
The probability of event $ R $ (rock music) and event $ N $ (Northern U.S.) is $ P(R\cap N)=\frac{\text{Number of people who like rock music and are from North}}{\text{Total number of people}} $.
## Step2: Identify values from table
From the table, the number of people who like rock music and are from North is $ 88 $, and the total number of people is $ 312 $.
## Step3: Calculate probability
So, $ P(R\cap N)=\frac{88}{312} $. Simplify the fraction: divide numerator and denominator by 8, we get $ \frac{11}{39} $.

# Answer: $\frac{11}{39}$

### Part c
# Explanation:
## Step1: Recall conditional probability formula
The conditional probability $ P(O|S)=\frac{P(O\cap S)}{P(S)}=\frac{\text{Number of people from South who like oldies}}{\text{Number of people from South}} $.
## Step2: Identify values from table
From the table, the number of people from South who like oldies is $ 44 $, and the number of people from South is $ 164 $.
## Step3: Calculate probability
So, $ P(O|S)=\frac{44}{164} $. Simplify the fraction: divide numerator and denominator by 4, we get $ \frac{11}{41} $.

# Answer: $\frac{11}{41}$

### Part d
# Explanation:
## Step1: Recall probability formula
$ P(R)=\frac{\text{Number of people who like rock music}}{\text{Total number of people}} $.
## Step2: Identify values from table
From the table, the number of people who like rock music is $ 138 $, and the total number of people is $ 312 $.
## Step3: Calculate and round
$ P(R)=\frac{138}{312}\approx0.44 $ (rounded to two decimal places).

# Answer: $ 0.44 $

### Part e
# Explanation:
## Step1: Recall probability formula
$ P(S)=\frac{\text{Number of people from South}}{\text{Total number of people}} $.
## Step2: Identify values from table
From the table, the number of people from South is $ 164 $, and the total number of people is $ 312 $.
## Step3: Calculate and round
$ P(S)=\frac{164}{312}\approx0.53 $ (rounded to two decimal places).

# Answer: $ 0.53 $

### Part f
# Explanation:
## Step1: Recall conditional probability formula
$ P(R|S)=\frac{P(R\cap S)}{P(S)}=\frac{\text{Number of people from South who like rock music}}{\text{Number of people from South}} $.
## Step2: Identify values from table
From the table, the number of people from South who like rock music is $ 50 $, and the number of people from South is $ 164 $. So, $ P(R|S)=\frac{50}{164}=\frac{25}{82}\approx0.30 $.
## Step3: Recall independence condition
Two events $ R $ and $ S $ are independent if $ P(R|S) = P(R) $. We know from part d that $ P(R)\approx0.44 $ and $ P(R|S)\approx0.30 $. Since $ 0.30
eq0.44 $, events $ R $ and $ S $ are associated (dependent) events.

# Answer: $ P(R|S)=\frac{25}{82}\approx0.30 $, and events $ R $ and $ S $ are associated events.

### Part g
# Explanation:
## Step1: Choose two events
Let's choose event $ A $: liking country music ( $ C $) and event $ B $: being from Northern U.S. ( $ N $).
## Step2: Calculate $ P(C|N) $
$ P(C|N)=\frac{\text{Number of people from North who like country music}}{\text{Number of people from North}}=\frac{11}{148}\approx0.07 $.
## Step3: Calculate $ P(N|C) $
$ P(N|C)=\frac{\text{Number of people who like country music and are from North}}{\text{Number of people who like country music}}=\frac{11}{81}\approx0.14 $.
## Step4: Compare the two probabilities
Since $ 0.07
eq0.14 $, $ P(A|B)
eq P(B|A) $. So Lisa is correct.

# Answer: Lisa is correct. For example, if $ A $ is liking country music and $ B $ is being from Northern U.S., $ P(A|B)=\frac{11}{148}\approx0.07 $ and $ P(B|A)=\frac{11}{81}\approx0.14 $, which are not equal.

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QUESTION IMAGE

Question

Explanation:

Part a

Step1: Recall probability formula

Step2: Identify values from table

Step3: Calculate probability

Step1: Recall probability formula for intersection

Step2: Identify values from table

Step3: Calculate probability

Step1: Recall conditional probability formula

Step2: Identify values from table

Step3: Calculate probability

Answer:

Part b