Musical styles other than rock and pop are becoming more popular. A survey of college students finds that 48% like country music, 30% like gospel music, and 10% like both. What is the conditional probability (±0.01) that a student likes gospel music, given that he or she likes country music
Conditional probability of A given B is given by
P(A|B)=P(A∩B)/P(B)
C=country, P(C)=0.48
G=gospel, P(G)=0.30
C∩G=both, P(C∩G)=0.10
P(gospel given country)
=P(G|C)
=P(G∩C)/P(C)
=0.10/0.48
=0.21
p(c)=0.48
p(g)=0.30
p(cng)=0.10
p(CuN)=0.48+0.30-0.10
=0.68
To find the conditional probability that a student likes gospel music given that they like country music, we need to use the formula for conditional probability:
P(Gospel | Country) = P(Gospel and Country) / P(Country)
First, we need to find the probability of liking both gospel and country music:
P(Gospel and Country) = 10% (given in the question)
Next, we need to find the probability of liking country music:
P(Country) = 48% (given in the question)
Now we can calculate the conditional probability:
P(Gospel | Country) = P(Gospel and Country) / P(Country)
P(Gospel | Country) = 10% / 48%
Calculating this:
P(Gospel | Country) ≈ 0.2083 (rounded to ±0.01)
Therefore, the conditional probability that a student likes gospel music given that he or she likes country music is approximately 0.2083 or 20.83% (rounded to ±0.01).
To find the conditional probability that a student likes gospel music given that he or she likes country music, you need to use the concept of conditional probability.
Conditional probability measures the probability of an event occurring given that another event has already occurred. In this case, we want to find the probability that a student likes gospel music, given that he or she already likes country music.
The conditional probability formula is:
P(A|B) = P(A and B) / P(B)
Where:
- P(A|B) is the conditional probability of event A occurring given that event B has already occurred.
- P(A and B) is the probability of both events A and B occurring.
- P(B) is the probability of event B occurring.
In this case, event A is liking gospel music, and event B is liking country music.
Given information:
- 48% of college students like country music.
- 30% of college students like gospel music.
- 10% of college students like both country music and gospel music.
To calculate P(A and B) (the probability of both events A and B occurring):
P(A and B) = P(B) * P(A|B)
P(A and B) = 0.48 * 0.10
P(A and B) = 0.048
To calculate P(B) (the probability of event B occurring):
P(B) = 0.48
Now, we can substitute the values into the conditional probability formula to find P(A|B):
P(A|B) = P(A and B) / P(B)
P(A|B) = 0.048 / 0.48
P(A|B) ≈ 0.1
Therefore, the conditional probability (±0.01) that a student likes gospel music, given that he or she likes country music, is approximately 0.1 or 10%.