https://file.aiquickdraw.com/mathbot/file/1725717915902syq9b932.jpg

can you explain the maths behind this thoroughly

The image you've provided appears to be a probability problem, likely using conditional probability. Let's go through the math step-by-step:

### Problem Breakdown:

1. **Context**: The problem appears to focus on kicking a ball at least 50 meters. It asks for the probability that at least three of the freshmen kick the ball at least 50 meters, **given** that the first chosen freshman kicks the ball at least 50 meters. This is a conditional probability question.

2. **Conditional Probability Formula**:
   The general formula for conditional probability is:

   \[
   P(A | B) = \frac{P(A \cap B)}{P(B)}
   \]
   Where:
   - \( P(A | B) \) is the probability of event \( A \) occurring given that event \( B \) has occurred.
   - \( P(A \cap B) \) is the probability that both \( A \) and \( B \) occur.
   - \( P(B) \) is the probability of event \( B \).

3. **Setup**: 
   - \( A \): At least 3 freshmen (including the first) kick the ball at least 50 m.
   - \( B \): The first freshman kicks the ball at least 50 m.

4. **Equation**:
   From the image, the formula is written as:

   \[
   P(X \geq 3 \, | \, \text{first}) = \frac{\text{first one and at least 2 more}}{\text{first one}}
   \]

   This matches the conditional probability formula. Now, let's break down the numerator and denominator:
   - The numerator is the probability that **the first freshman kicks the ball at least 50 meters**, and **at least two more freshmen** also kick the ball at least 50 meters.
   - The denominator is the probability that **the first freshman kicks the ball at least 50 meters**.

5. **Numerical Values**:
   - The probability of any given freshman kicking the ball at least 50 meters seems to be 0.7 (based on context).
   
   The formula given in the image continues as:
   \[
   P(X \geq 3 \, | \, \text{first}) = \frac{0.7 \times \text{Bin}( \text{at least 2 of 5} )}{0.7}
   \]
   Here:
   - \( 0.7 \) represents the probability that the first freshman kicks the ball at least 50 m.
   - \( \text{Bin(at least 2 of 5)} \) represents a binomial probability of **at least 2 out of the remaining 5 freshmen** kicking the ball at least 50 m.

6. **Simplification**:
   The \( 0.7 \) terms in the numerator and denominator cancel out, so the equation simplifies to:

   \[
   P(X \geq 3 \, | \, \text{first}) = \text{Bin(at least 2 of 5)} = 0.323
   \]

   This final value \( 0.323 \) is the probability that **at least 3 freshmen** (including the first) kick the ball at least 50 meters.

### Key Concepts:
- **Binomial Probability**: In the simplified formula, the remaining probability (after canceling out the \( 0.7 \)) involves a binomial probability. This represents the chance of having **at least 2 successes out of 5 independent trials**, where the probability of success on each trial is 0.7.
- **Conditional Probability**: The structure of the problem revolves around finding the probability of a compound event given a specific condition.

Would you like further elaboration on any specific part? Below are some questions to expand your understanding.

---

### Related Questions:
1. How do you calculate binomial probabilities for different values of successes?
2. What is the difference between independent and conditional probability?
3. How does simplifying terms like \( 0.7 \) on both sides affect the calculation?
4. What does the "at least 2 out of 5" binomial part represent in detail?
5. Can you generalize this method for any probability condition involving more trials?

**Tip**: Always check if you can simplify terms when using conditional probability; it makes complex problems easier to solve.

Can you explain the maths behind this thoroughly?

Explore the detailed solution to a probability problem involving conditional and binomial probability. Learn how to calculate the probability that at least three freshmen kick the ball at least 50 meters, given that the first chosen freshman kicks the ball at least 50 meters. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation