In a study of helicopter usage and patient​ survival, among the 57 comma 749 patients transported by​ helicopter, 243 of them left the treatment center against medical​ advice, and the other 57 comma 506 did not leave against medical advice. If 50 of the subjects transported by helicopter are randomly selected without​ replacement, what is the probability that none of them left the treatment center against medical​ advice?
Question content area bottom
Part 1
The probability is 
  


To solve this problem, we'll calculate the probability that none of the 50 randomly selected patients left the treatment center against medical advice. This is a problem of selecting without replacement, so we will use the hypergeometric distribution.

### Step-by-Step Solution:

1. **Total number of patients transported by helicopter**:  
   \( N = 57,749 \)

2. **Number of patients who left against medical advice**:  
   \( M = 243 \)

3. **Number of patients who did not leave against medical advice**:  
   \( N - M = 57,506 \)

4. **Number of patients randomly selected**:  
   \( n = 50 \)

We need to calculate the probability that all 50 selected patients did **not** leave against medical advice.

### Hypergeometric Probability:
The probability of selecting \( k \) patients who did not leave against medical advice in a sample of size \( n \) is given by the hypergeometric distribution:

\[
P(\text{none left against medical advice}) = \frac{\binom{57,506}{50} \times \binom{243}{0}}{\binom{57,749}{50}}
\]

Since \(\binom{243}{0} = 1\), the formula simplifies to:

\[
P(\text{none left against medical advice}) = \frac{\binom{57,506}{50}}{\binom{57,749}{50}}
\]

### Approximation:
For large numbers, the hypergeometric distribution can be approximated by:

\[
P(\text{none left against medical advice}) \approx \frac{57,506}{57,749} \times \frac{57,505}{57,748} \times \cdots \times \frac{57,457}{57,700}
\]

This can be computed as:

\[
P(\text{none left against medical advice}) \approx \prod_{i=0}^{49} \frac{57,506 - i}{57,749 - i}
\]

### Calculation:
Using this approximation or a calculator, we find:

\[
P(\text{none left against medical advice}) \approx \frac{57,506}{57,749} \times \frac{57,505}{57,748} \times \ldots \times \frac{57,457}{57,700}
\]

The actual computation for this would result in:

\[
P(\text{none left against medical advice

In a study of helicopter usage and patient survival, among the 57,749 patients transported by helicopter, 243 of them left the treatment center against medical advice, and the other 57,506 did not leave against medical advice. If 50 of the subjects transported by helicopter are randomly selected without replacement, what is the probability that none of them left the treatment center against medical advice?

Calculate the probability that none of 50 randomly selected patients left the treatment center against medical advice, using the hypergeometric distribution. This problem involves understanding probability concepts and is suitable for undergraduate 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