Math Problem Statement

In a class of 150 students, 96 are taking math (M), 79 are taking science (S), and 54 are taking both math and science. One student is picked at random. Find the probability of taking math or science or both.

Solution

Problem Analysis

The question asks for the probability that a randomly selected student is taking math, science, or both. This requires calculating the union of two sets: MM (students taking math) and SS (students taking science). The formula for the union of two sets is:

P(MS)=P(M)+P(S)P(MS)P(M \cup S) = P(M) + P(S) - P(M \cap S)

Given Information:

  • Total students: N=150N = 150
  • Students taking math (MM): M=96|M| = 96
  • Students taking science (SS): S=79|S| = 79
  • Students taking both math and science (MSM \cap S): MS=54|M \cap S| = 54

Calculation:

We need to calculate P(MS)P(M \cup S), the probability of a student being in at least one of the sets (math, science, or both). Using the formula:

P(MS)=MN+SNMSNP(M \cup S) = \frac{|M|}{N} + \frac{|S|}{N} - \frac{|M \cap S|}{N}

Substitute the values:

P(MS)=96150+7915054150P(M \cup S) = \frac{96}{150} + \frac{79}{150} - \frac{54}{150}

Step-by-step Computation:

  1. 96150=0.64\frac{96}{150} = 0.64
  2. 79150=0.5267\frac{79}{150} = 0.5267
  3. 54150=0.36\frac{54}{150} = 0.36

P(MS)=0.64+0.52670.36=0.8067P(M \cup S) = 0.64 + 0.5267 - 0.36 = 0.8067

Final Answer:

P(MS)=0.807(rounded to three decimal places)P(M \cup S) = 0.807 \quad (\text{rounded to three decimal places})

Would you like further explanation or additional details?


Related Questions:

  1. How do you calculate the probability of the intersection of two sets (MSM \cap S)?
  2. What is the probability that a randomly selected student is taking only math?
  3. What is the probability that a student is taking only science?
  4. If a student is known to be taking math, what is the probability they are also taking science?
  5. How would the result change if the total number of students was different?

Tip:

When dealing with union and intersection in probability, always ensure you account for overlapping elements to avoid double-counting!

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory
Union of Sets

Formulas

P(M ∪ S) = P(M) + P(S) - P(M ∩ S)

Theorems

Basic Probability Theorem

Suitable Grade Level

Grades 9-12