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?
$\frac{3}{10}$
$\frac{3}{4}$
$\frac{9}{10}$
$\frac{1}{10}$
Answer (2)
Define sets for music (M) and dance (D) students.
Calculate the number of students in M ∪ D : 100 − 30 = 70 .
Use the formula n ( M ∪ D ) = n ( M ) + n ( D ) − n ( M ∩ D ) to find n ( M ∩ D ) = 10 .
Calculate the probability: 100 10 = 10 1 .
Explanation
Analyze the problem Let's analyze the problem. We have information about students taking music classes, dance classes, and those taking neither. We need to find the probability that a student took both music and dance classes.
Define sets and notations Let M be the set of students who took music classes, and D be the set of students who took dance classes. Let n ( M ) be the number of students who took music classes, n ( D ) be the number of students who took dance classes, and n ( M ∪ D ) be the number of students who took either music or dance classes or both.
Calculate the number of students in M union D We are given: n ( M ) = 35 n ( D ) = 45 Number of students who took neither music nor dance = 30 Total number of students = 100 Therefore, the number of students who took either music or dance or both is: n ( M ∪ D ) = 100 − 30 = 70
State the formula for the union of two sets We want to find the number of students who took both music and dance classes, which is n ( M ∩ D ) .
We can use the formula for the union of two sets: n ( M ∪ D ) = n ( M ) + n ( D ) − n ( M ∩ D )
Calculate the number of students in M intersection D Plugging in the given values, we have: 70 = 35 + 45 − n ( M ∩ D ) Solving for n ( M ∩ D ) , we get: n ( M ∩ D ) = 35 + 45 − 70 = 80 − 70 = 10
Calculate the probability The probability that a randomly chosen student took both music and dance classes is: Total number of students n ( M ∩ D ) = 100 10 = 10 1
Examples
Imagine you're organizing after-school activities and want to know the likelihood that a student is interested in both music and dance. This probability helps you plan combined sessions or allocate resources effectively, ensuring you cater to students with diverse interests.
The probability that a randomly chosen student took both music and dance classes is 10 1 . This was calculated using the total number of students and their participation in each class. Using set theory, we determined the number of students taking both classes and extracted the probability from that.
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