A test is given to a very large (>1000) group of students at a school in Minneapolis. We obtain information on the number of students who received a certain score, as well as what eye color they have. Count information is presented in a table similar to the one below.

A

C
D
Total
Blue Eyes

Brown Eyes

Green Eyes

Black Eyes

Total

Administrators of the Minneapolis school are trying to find the values of two probabilities:
1. Find the probability that one randomly selected student has black eyes AND got a "C".
2. Find the probability that one randomly selected student has black eyes, GIVEN they got a "C".
Your Questions:
You MUST do this before you submit your exam!!
1. Describe the difference between the two questions the administrators are asking.
2. Describe the difference in how you would calculate the probabilities.

The two probabilities the administrators are asking for are different types of probabilities: one is a **joint probability** and the other is a **conditional probability**. Let’s break down the differences between the two questions.

### 1. **Difference between the two questions:**

- **Question 1 (Joint Probability):** The first question asks for the probability that a randomly selected student both **has black eyes** and **received a "C"**. This is the probability of two events happening simultaneously.
  
  - **P(Black Eyes and C)**: This is the probability that a randomly chosen student satisfies both conditions — having black eyes and scoring a C. This is a joint probability.

- **Question 2 (Conditional Probability):** The second question asks for the probability that a randomly selected student has black eyes, **given** that they already received a "C". This is the probability of having black eyes, given that we know the student scored a "C".

  - **P(Black Eyes | C)**: This is the probability of having black eyes, under the condition (or assumption) that the student has already been determined to have scored a C. This is a conditional probability.

### 2. **Difference in how you would calculate the probabilities:**

- **For the Joint Probability** (P(Black Eyes and C)):  
  You calculate this by dividing the number of students who have both **black eyes** and received a **C** by the **total number of students**.
  \[
  P(Black Eyes \text{ and } C) = \frac{\text{Number of students with black eyes and received a "C"}}{\text{Total number of students}}.
  \]

- **For the Conditional Probability** (P(Black Eyes | C)):  
  You calculate this by dividing the number of students who have both **black eyes** and received a **C** by the **total number of students who received a "C"**.
  \[
  P(Black Eyes | C) = \frac{\text{Number of students with black eyes and received a "C"}}{\text{Total number of students who received a "C"}}.
  \]

### Summary of differences:

- **Joint probability** is concerned with the likelihood of two events occurring together, while **conditional probability** is focused on the likelihood of one event occurring given that another event has already occurred.
- The **joint probability** divides by the total number of students, while the **conditional probability** divides by the number of students who received a "C".

### Do you have any questions on this, or would you like me to expand on any particular part?

---

Here are five related questions you could explore:
1. How would these probabilities change if the number of students in each category was known?
2. What would the joint probability be if no students with black eyes received a C?
3. How does the formula for conditional probability change if we wanted P(C | Black Eyes)?
4. Could you calculate the probability of having brown eyes given that a student got a D?
5. What assumptions are made when calculating these probabilities in real data collection?

**Tip:** Joint and conditional probabilities are related; if you know one, you can often calculate the other using the formula:  
\[ P(A | B) = \frac{P(A \text{ and } B)}{P(B)}. \]

A test is given to a very large (>1000) group of students at a school in Minneapolis. We obtain information on the number of students who received a certain score, as well as what eye color they have. Count information is presented in a table similar to the one below.

Administrators of the Minneapolis school are trying to find the values of two probabilities:
1. Find the probability that one randomly selected student has black eyes AND got a "C".
2. Find the probability that one randomly selected student has black eyes, GIVEN they got a "C".

You MUST do this before you submit your exam!!
1. Describe the difference between the two questions the administrators are asking.
2. Describe the difference in how you would calculate the probabilities.

Explore how to calculate joint and conditional probabilities using a real-world example of student scores and eye colors in a school. Learn the difference between the probability of two events happening together (black eyes and receiving a 'C') and the probability of one event given the other. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation