Math / Data & Statistics

Question5. The following table shows music preferences found by a survey of the faculty at a local university. Express your answers in fraction form. \begin{tabular}{|l|c|c|c|c|} \hline & Country Music (C) & Rock Music (R) & Oldies (O) & Total \\ \hline Northern U.S. (N) & 11 & 88 & 49 & 148 \\ \hline Southern U.S. (S) & 70 & 50 & 44 & 164 \\ \hline Total & 81 & 138 & 93 & 312 \\ \hline \end{tabular} 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 $P(R \mid S)$ 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 $P(A \mid B)=P(B \mid A)$ for any two events $A$ and $B$ . Lisa says that $P(A \mid B)$ and $P(B \mid A)$ do not have to be equal. Pick two events from the table in problem 5 and determine who is correct.

Studdy Solution

STEP 1

1. The table provides the number of faculty members with specific music preferences from different regions.
2. Probabilities are calculated as the ratio of favorable outcomes to the total number of outcomes.
3. Conditional probability is calculated using the formula $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ .

STEP 2

1. Calculate the probability of liking country music.
2. Calculate the probability of liking rock music and being from the North.
3. Calculate the conditional probability of liking oldies given being from the South.
4. Calculate the probability of liking rock music in decimal form.
5. Calculate the probability of being from the South in decimal form.
6. Calculate the conditional probability of liking rock music given being from the South and determine independence.
7. Evaluate Zoe and Lisa's disagreement using conditional probabilities from the table.

STEP 3

Calculate the probability of liking country music:
$P(C) = \frac{\text{Total liking Country Music}}{\text{Total surveyed}} = \frac{81}{312}$

STEP 4

Calculate the probability of liking rock music and being from the North:
$P(R \cap N) = \frac{\text{Northern U.S. liking Rock Music}}{\text{Total surveyed}} = \frac{88}{312}$

STEP 5

Calculate the conditional probability of liking oldies given being from the South:
$P(O \mid S) = \frac{\text{Southern U.S. liking Oldies}}{\text{Total Southern U.S.}} = \frac{44}{164}$

STEP 6

Calculate the probability of liking rock music in decimal form:
$P(R) = \frac{\text{Total liking Rock Music}}{\text{Total surveyed}} = \frac{138}{312} \approx 0.44$

STEP 7

Calculate the probability of being from the South in decimal form:
$P(S) = \frac{\text{Total Southern U.S.}}{\text{Total surveyed}} = \frac{164}{312} \approx 0.53$

STEP 8

Calculate the conditional probability of liking rock music given being from the South:
$P(R \mid S) = \frac{\text{Southern U.S. liking Rock Music}}{\text{Total Southern U.S.}} = \frac{50}{164}$
Determine independence:
Events $R$ and $S$ are independent if $P(R \mid S) = P(R)$ . Since $P(R \mid S) \neq P(R)$ , events $R$ and $S$ are associated.

STEP 9

Evaluate Zoe and Lisa's disagreement:
Choose events $A = \text{liking Country Music}$ and $B = \text{being from the North}$ .
$P(A \mid B) = \frac{11}{148}$ $P(B \mid A) = \frac{11}{81}$
Since $P(A \mid B) \neq P(B \mid A)$ , Lisa is correct.
The solutions are: a. $\frac{81}{312}$ b. $\frac{88}{312}$ c. $\frac{44}{164}$ d. $0.44$ e. $0.53$ f. $\frac{50}{164}$ , events are associated g. Lisa is correct

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Studdy Solution

STEP 1

STEP 2

STEP 3

Calculate the probability of liking country music:
$P(C) = \frac{\text{Total liking Country Music}}{\text{Total surveyed}} = \frac{81}{312}$

STEP 4

Calculate the probability of liking rock music and being from the North:
$P(R \cap N) = \frac{\text{Northern U.S. liking Rock Music}}{\text{Total surveyed}} = \frac{88}{312}$

STEP 5

Calculate the conditional probability of liking oldies given being from the South:
$P(O \mid S) = \frac{\text{Southern U.S. liking Oldies}}{\text{Total Southern U.S.}} = \frac{44}{164}$

STEP 6

Calculate the probability of liking rock music in decimal form:
$P(R) = \frac{\text{Total liking Rock Music}}{\text{Total surveyed}} = \frac{138}{312} \approx 0.44$

STEP 7

Calculate the probability of being from the South in decimal form:
$P(S) = \frac{\text{Total Southern U.S.}}{\text{Total surveyed}} = \frac{164}{312} \approx 0.53$

STEP 8

STEP 9

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Studdy Solution

STEP 1

STEP 2

STEP 3

Calculate the probability of liking country music:P(C)=Total liking Country MusicTotal surveyed=81312 P(C) = \frac{\text{Total liking Country Music}}{\text{Total surveyed}} = \frac{81}{312} P(C)=Total surveyedTotal liking Country Music​=31281​

STEP 4

Calculate the probability of liking rock music and being from the North:P(R∩N)=Northern U.S. liking Rock MusicTotal surveyed=88312 P(R \cap N) = \frac{\text{Northern U.S. liking Rock Music}}{\text{Total surveyed}} = \frac{88}{312} P(R∩N)=Total surveyedNorthern U.S. liking Rock Music​=31288​

STEP 5

STEP 6

Calculate the probability of liking rock music in decimal form:P(R)=Total liking Rock MusicTotal surveyed=138312≈0.44 P(R) = \frac{\text{Total liking Rock Music}}{\text{Total surveyed}} = \frac{138}{312} \approx 0.44 P(R)=Total surveyedTotal liking Rock Music​=312138​≈0.44

STEP 7

Calculate the probability of being from the South in decimal form:P(S)=Total Southern U.S.Total surveyed=164312≈0.53 P(S) = \frac{\text{Total Southern U.S.}}{\text{Total surveyed}} = \frac{164}{312} \approx 0.53 P(S)=Total surveyedTotal Southern U.S.​=312164​≈0.53

STEP 8

STEP 9

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Calculate the probability of liking country music:
$P(C) = \frac{\text{Total liking Country Music}}{\text{Total surveyed}} = \frac{81}{312}$

Calculate the probability of liking rock music and being from the North:
$P(R \cap N) = \frac{\text{Northern U.S. liking Rock Music}}{\text{Total surveyed}} = \frac{88}{312}$

Calculate the probability of liking rock music in decimal form:
$P(R) = \frac{\text{Total liking Rock Music}}{\text{Total surveyed}} = \frac{138}{312} \approx 0.44$

Calculate the probability of being from the South in decimal form:
$P(S) = \frac{\text{Total Southern U.S.}}{\text{Total surveyed}} = \frac{164}{312} \approx 0.53$