Math Problem Statement

A student is randomly selected from this table. What is the probability that they are a boy, given that they are a freshman?

Solution

The problem is asking for the conditional probability that a student is a boy, given that they are a freshman.

To solve this, we use the formula for conditional probability:

P(Boy | Freshman)=P(Boy and Freshman)P(Freshman)P(\text{Boy | Freshman}) = \frac{P(\text{Boy and Freshman})}{P(\text{Freshman})}

Step 1: Find P(Boy and Freshman)P(\text{Boy and Freshman})

From the table, the number of boys who are freshmen is 7.

Step 2: Find P(Freshman)P(\text{Freshman})

The total number of freshmen (both boys and girls) is: 7 (boys)+5 (girls)=127 \text{ (boys)} + 5 \text{ (girls)} = 12

Step 3: Compute the conditional probability

P(Boy | Freshman)=712P(\text{Boy | Freshman}) = \frac{7}{12}

Thus, the probability that a student is a boy, given that they are a freshman, is:

P(Boy | Freshman)=7120.5833P(\text{Boy | Freshman}) = \frac{7}{12} \approx 0.5833

Would you like more details on this solution or have any questions?

Here are some related questions you might explore:

  1. What is the probability that a student is a girl, given that they are a freshman?
  2. What is the total probability that a randomly selected student is a boy?
  3. What is the probability that a student is a junior, given they are a boy?
  4. What is the probability that a student is a sophomore, given that they are a girl?
  5. How would the result change if more freshmen boys were added?

Tip: Conditional probability helps focus on a specific subgroup within the overall sample, making it useful in real-world situations where additional information is available.

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Math Problem Analysis

Mathematical Concepts

Conditional Probability
Probability

Formulas

P(Boy | Freshman) = P(Boy and Freshman) / P(Freshman)

Theorems

Conditional Probability Formula

Suitable Grade Level

Grades 9-12