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!