Question

3
at a music academy, a randomly selected group of people were asked which type of musical instrument they are learning to play. the results are displayed, by age, in the two - way frequency table.

	string	woodwind	piano	total
ages 20 - 24	9	10	17	36
ages 25 - 29	16	6	7	29
total	36	19	31	86

based on the data in the table, which statement is true?
a. $p(playing piano|ages 25 - 29)=p(ages 15 - 19|learning piano)$
b. $p(ages 15 - 19|learning piano)\
eq p(ages 15 - 19)$
c. $p(learning piano|ages 20 - 24)=p(learning piano)$
d. $p(learning a string instrument)\
eq p(ages 20 - 24)$

Explanation:

Step1: Recall conditional probability formula

The formula for conditional probability is \( P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)} \), where \( n(A\cap B) \) is the number of elements in both \( A \) and \( B \), and \( n(B) \) is the number of elements in \( B \). We will calculate the probabilities for each option.

Step2: Calculate probabilities for Option A

• Calculate \( P(\text{playing piano}|\text{ages 25 - 29}) \):
• \( n(\text{playing piano} \cap \text{ages 25 - 29}) = 7 \) (number of people aged 25 - 29 learning piano)
• \( n(\text{ages 25 - 29}) = 29 \) (total number of people aged 25 - 29)
• So \( P(\text{playing piano}|\text{ages 25 - 29})=\frac{7}{29}\approx0.241 \)
• Calculate \( P(\text{ages 15 - 19}|\text{learning piano}) \):
• \( n(\text{ages 15 - 19} \cap \text{learning piano}) = 7 \) (number of people aged 15 - 19 learning piano)
• \( n(\text{learning piano}) = 31 \) (total number of people learning piano)
• So \( P(\text{ages 15 - 19}|\text{learning piano})=\frac{7}{31}\approx0.226 \)
• Since \( \frac{7}{29}

eq\frac{7}{31} \), Option A is false.

Step3: Calculate probabilities for Option B

• Calculate \( P(\text{ages 15 - 19}|\text{learning piano}) \):
• As above, \( P(\text{ages 15 - 19}|\text{learning piano})=\frac{7}{31}\approx0.226 \)
• Calculate \( P(\text{ages 15 - 19}) \):
• \( n(\text{ages 15 - 19}) = 21 \), \( n(\text{total}) = 86 \)
• So \( P(\text{ages 15 - 19})=\frac{21}{86}\approx0.244 \)
• Since \( \frac{7}{31}\approx0.226

eq\frac{21}{86}\approx0.244 \), \( P(\text{ages 15 - 19}|\text{learning piano})
eq P(\text{ages 15 - 19}) \), Option B is true (but we need to check other options as well).

Step4: Calculate probabilities for Option C

• Calculate \( P(\text{learning piano}|\text{ages 20 - 24}) \):
• \( n(\text{learning piano} \cap \text{ages 20 - 24}) = 17 \) (number of people aged 20 - 24 learning piano)
• \( n(\text{ages 20 - 24}) = 36 \) (total number of people aged 20 - 24)
• So \( P(\text{learning piano}|\text{ages 20 - 24})=\frac{17}{36}\approx0.472 \)
• Calculate \( P(\text{learning piano}) \):
• \( n(\text{learning piano}) = 31 \), \( n(\text{total}) = 86 \)
• So \( P(\text{learning piano})=\frac{31}{86}\approx0.360 \)
• Since \( \frac{17}{36}\approx0.472

eq\frac{31}{86}\approx0.360 \), Option C is false.

Step5: Calculate probabilities for Option D

• Calculate \( P(\text{learning a string instrument}) \):
• \( n(\text{learning a string instrument}) = 36 \), \( n(\text{total}) = 86 \)
• So \( P(\text{learning a string instrument})=\frac{36}{86}\approx0.419 \)
• Calculate \( P(\text{ages 20 - 24}) \):
• \( n(\text{ages 20 - 24}) = 36 \), \( n(\text{total}) = 86 \)
• So \( P(\text{ages 20 - 24})=\frac{36}{86}\approx0.419 \)
• Since \( \frac{36}{86}=\frac{36}{86} \), Option D is false.

Answer:

B. \( P(\text{ages 15-19|learning piano})
eq P(\text{ages 15-19}) \)