Questions: The gender and year of each of the 216 students in a high school band were recorded. The data are summarized in the table below. Male: Freshman 41, Sophomore 31, Junior 22, Senior 11 Female: Freshman 30, Sophomore 37, Junior 26, Senior 18 Suppose a student from the high school band is chosen at random. (a) What is the probability that the student is male, given that the student is a senior? (b) What is the probability that the student is female or a sophomore?

Solution Steps

For part (a), we need to find the probability that the student is male, given that the student is a senior. From the table:

• Number of male seniors = 11
• Number of female seniors = 18
• Total number of seniors = \( 11 + 18 = 29 \)

The probability that the student is male, given that the student is a senior, is calculated using the formula for conditional probability: \[ P(\text{Male} \mid \text{Senior}) = \frac{\text{Number of male seniors}}{\text{Total number of seniors}} = \frac{11}{29} \] \[ P(\text{Male} \mid \text{Senior}) \approx 0.38 \]

For part (b), we need to find the probability that the student is female or a sophomore. From the table:

• Number of females = \( 30 + 37 + 26 + 18 = 111 \)
• Number of sophomores = \( 31 + 37 = 68 \)
• Number of female sophomores = 37

The probability that the student is female or a sophomore is calculated using the formula for the union of two events: \[ P(\text{Female} \cup \text{Sophomore}) = P(\text{Female}) + P(\text{Sophomore}) - P(\text{Female} \cap \text{Sophomore}) \] \[ P(\text{Female}) = \frac{111}{216}, \quad P(\text{Sophomore}) = \frac{68}{216}, \quad P(\text{Female} \cap \text{Sophomore}) = \frac{37}{216} \] \[ P(\text{Female} \cup \text{Sophomore}) = \frac{111}{216} + \frac{68}{216} - \frac{37}{216} = \frac{142}{216} \] \[ P(\text{Female} \cup \text{Sophomore}) \approx 0.66 \]

Final Answer