https://file.aiquickdraw.com/mathbot/file/1728245063221rrk8zobh.jpg

To analyze the graph of the probability distribution and solve the given problems, I'll need to carefully interpret the graph and its underlying distribution. Let's break it down:

### Interpretation of the graph:
- The graph represents a probability distribution, which indicates that the area under the curve represents the total probability (which must sum to 1).
- The curve is made of two curved segments and a flat line segment between \( x = 4 \) and \( x = 13 \), at a height of \( y = 0.075 \).
  
### Part (a):
We are asked to find the probability that the number picked is **not** between 4 and 13.

1. The probability of a number being between 4 and 13 corresponds to the area under the flat segment of the graph from \( x = 4 \) to \( x = 13 \). The length of this interval is:
   \[
   13 - 4 = 9
   \]
2. Since the height of the line is \( y = 0.075 \), the area (and hence the probability) is:
   \[
   P(4 \leq x \leq 13) = 9 \times 0.075 = 0.675
   \]
3. The probability that the number is **not** between 4 and 13 is the complement of this, which is:
   \[
   P(x \notin [4, 13]) = 1 - 0.675 = 0.325
   \]

### Part (b):
We are asked to find the probability that the number picked is between 6.2 and 9.6. This range lies within the flat section between \( x = 4 \) and \( x = 13 \), where the probability density is constant at \( 0.075 \).

1. The length of the interval from 6.2 to 9.6 is:
   \[
   9.6 - 6.2 = 3.4
   \]
2. The probability is the area under the curve for this interval:
   \[
   P(6.2 \leq x \leq 9.6) = 3.4 \times 0.075 = 0.255
   \]

### Part (c):
We are asked to find the probability that the number picked is exactly 13.

1. In a continuous probability distribution, the probability of picking any exact value (such as \( x = 13 \)) is zero, because the area under a single point is infinitesimal. Therefore:
   \[
   P(x = 13) = 0
   \]

### Summary of Results:
- (a) Probability that the number picked is **not** between 4 and 13: \( 0.325 \)
- (b) Probability that the number picked is between 6.2 and 9.6: \( 0.255 \)
- (c) Probability that the number picked is exactly 13: \( 0 \)

Let me know if you would like more details on any part or explanations of the integration! Here's a few related questions for further exploration:

1. What is the total area under the curve for \( x = 0 \) to \( x = 17 \)?
2. How would the probability change if the distribution height between 4 and 13 was altered?
3. What is the shape of the curve outside of \( x = 4 \) and \( x = 13 \)? 
4. How does the probability of selecting a number between 0 and 4 compare to that between 13 and 17?
5. How would you find the median of this distribution?

**Tip:** In continuous probability distributions, the probability of selecting an exact value (like in part c) is always zero!

Find the probability that the number picked is not between 4 and 13, between 6.2 and 9.6, and exactly 13.

Solve a probability distribution problem involving intervals between 4 and 13, 6.2 and 9.6, and the exact point of 13. The solution includes using the area under the curve and the complement rule, 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