Question

select the correct answer. during the summer vacation, 35 students out of 100 took up music classes, 45 students took up dance classes, and 30 students did not take up either music or dance classes. if a student is chosen at random, what is the probability that the student took both music and dance classes?

Explanation:

Step1: Find number of students who took both classes

The number of students who took music is 35, the number of students who took dance is 45, and the number of students who took neither is 30. The total number of students is 100. Let \(M\) be the set of music - taking students and \(D\) be the set of dance - taking students. We know that \(n(M\cup D)=100 - 30=70\) (the number of students who took either music or dance or both). By the formula \(n(M\cup D)=n(M)+n(D)-n(M\cap D)\), we substitute \(n(M) = 35\), \(n(D)=45\) and \(n(M\cup D)=70\) into it. So \(70=35 + 45-n(M\cap D)\).

Step2: Solve for \(n(M\cap D)\)

Rearranging the equation \(70=35 + 45-n(M\cap D)\) gives \(n(M\cap D)=35 + 45-70=10\).

Step3: Calculate the probability

The probability \(P\) that a randomly - chosen student took both music and dance is given by the formula \(P=\frac{n(M\cap D)}{n(\text{total})}\). Since \(n(M\cap D) = 10\) and \(n(\text{total})=100\), then \(P=\frac{10}{100}=\frac{1}{10}\).

Answer:

\(\frac{1}{10}\)