To solve the problem, let's analyze it in parts.

We are given a probability distribution over the interval $[2, 9]$, with the area of regions $A$, $B$, and $C$ corresponding to the probabilities associated with those segments. Here’s the breakdown:

• Region $A$: From $2$ to $5$ with an area of $0.30$
• Region $B$: From $5$ to $7$ with an area of $0.45$
• Region $C$: From $7$ to $9$ with an area of $0.25$

The areas represent the probabilities of picking a number within each respective region.

Part (a): Find the probability that a real number between $5$ and $7$ is picked.

Since the interval from $5$ to $7$ corresponds exactly to Region $B$, we can directly use the area of $B$ for the probability.

$\text{Probability (5 to 7)} = \text{Area of Region } B = 0.45$

Part (b): Find the probability that a real number between $5$ and $9$ is picked.

The interval from $5$ to $9$ includes both Region $B$ and Region $C$. Therefore, we add the areas of both regions to get the probability.

$\text{Probability (5 to 9)} = \text{Area of Region } B + \text{Area of Region } C = 0.45 + 0.25 = 0.70$

Final Answers:

• (a) $0.45$
• (b) $0.70$

Would you like further explanation on any part of this solution?

Related Questions:

1. What is the total probability over the entire interval $[2, 9]$?
2. How would the probabilities change if the areas of regions $A$, $B$, and $C$ were different?
3. If Region $B$ had a smaller area, how would that impact the probability for interval $5$ to $7$?
4. Can you find the probability of picking a number in the interval $3$ to $7$?
5. How would you represent this probability distribution graphically if it wasn’t already provided?

Tip:

Remember, in a probability distribution over a continuous interval, the area under the curve represents the probability of choosing a value within that interval.