Question

in a group of 90 students, 50 are in band, 35 are in choir, and 12 are in both. what is the probability of a student being in band or choir? p(band or choir) = ?%

Explanation:

Step1: Use the principle of inclusion - exclusion

The formula for \( P(A\cup B) \) is \( n(A)+n(B)-n(A\cap B) \), where \( n(A) \) is the number of elements in set \( A \), \( n(B) \) is the number of elements in set \( B \), and \( n(A\cap B) \) is the number of elements in the intersection of \( A \) and \( B \). Here, \( n(\text{Band}) = 50 \), \( n(\text{Choir})=35 \), and \( n(\text{Band}\cap\text{Choir}) = 12 \). So the number of students in band or choir is \( 50 + 35-12=73 \).

Step2: Calculate the probability

The total number of students is \( 90 \). The probability \( P(\text{Band or Choir})=\frac{73}{90}\approx0.8111 \). To convert to a percentage, we multiply by \( 100 \), so \( 0.8111\times100 = 81.11\% \) (rounded to two decimal places).

Answer:

\( 81.11\% \) (or if we use exact fraction, \( \frac{73}{90}\times100=\frac{7300}{90}\approx81.11\% \))