a. The conditional probability of being on the marching band, given that the student plays a team sport, is approximately 0.0171.
b. The probability of being on the marching band, regardless of team sport participation, is approximately 0.0521.
c. The events "on the marching band" and "on a team sport" are associated since they are dependent events.
How to calculate conditional probability
To answer the questions, use the given data and apply the concepts of conditional probability and independence of events.
a. To find the conditional probability of being on the marching band, given that the student plays a team sport, we need to calculate P(Marching band | Team sport). This can be done using the formula:
P(Marching band | Team sport) = P(Marching band and Team sport) / P(Team sport)
From the given data:
P(Marching band and Team sport) = 14
P(Team sport) = 818
Therefore,
P(Marching band | Team sport) = 14 / 818 ≈ 0.0171
The conditional probability of being on the marching band, given that the student plays a team sport, is approximately 0.0171.
b. The probability of being on the marching band refers to the overall probability of a student being in the marching band, regardless of whether they play a team sport or not.
This can be calculated using the formula:
P(Marching band) = (Number of students on the marching band) / (Total number of students)
From the given data:
Number of students on the marching band = 222
Total number of students = 222 + 1909 + 1313 + 818 = 4262
Therefore,
P(Marching band) = 222 / 4262 ≈ 0.0521
The probability of being on the marching band is approximately 0.0521.
The difference between part (a) and part (b) is that part (a) calculates the conditional probability of being on the marching band among students who play a team sport, while part (b) calculates the overall probability of being on the marching band, regardless of team sport participation.
c. To determine if the events "on the marching band" and "on a team sport" are associated, compare their probabilities.
If the events are independent, the probability of both events occurring should be equal to the product of their individual probabilities.
P(Marching band and Team sport) = P(Marching band) * P(Team sport)
From the given data:
P(Marching band and Team sport) = 14
P(Marching band) ≈ 0.0521
P(Team sport) = 818 / 4262 ≈ 0.1919
Calculating the product of the individual probabilities:
P(Marching band) * P(Team sport) ≈ 0.0521 * 0.1919 ≈ 0.00999
Since the product of the individual probabilities is not equal to the joint probability, the events "on the marching band" and "on a team sport" are dependent events. Therefore, they are associated.