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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.

A real number between 2 and 9 will be picked according to the probability distribution shown in the figure. Regions under the curve are labeled with A, B, and C. The area of each region is shown in the table. Use the figure and the table to answer the parts below.
(a) Find the probability that a real number between 5 and 7 is picked.
(b) Find the probability that a real number between 5 and 9 is picked.

Learn how to calculate probabilities using areas under a continuous probability distribution curve. Given intervals [2, 5], [5, 7], and [7, 9] with respective areas of 0.30, 0.45, and 0.25, find the probabilities for intervals [5, 7] and [5, 9]. Ideal for students in Grades 10-12.

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