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Let's break down the problem step by step.

### Part (a): Find the probability distribution for the sum of the two slips.

You have four slips numbered 22, 23, 24, and 25. When you draw two slips, the possible sums are:
- \(22 + 23 = 45\)
- \(22 + 24 = 46\)
- \(22 + 25 = 47\)
- \(23 + 24 = 47\)
- \(23 + 25 = 48\)
- \(24 + 25 = 49\)

Now, the probability of drawing any specific pair of slips is equally likely. There are 6 possible outcomes (since you are drawing two slips without replacement), and each sum appears with the following frequency:
- Sum 45 appears once: \( P(45) = \frac{1}{6} \)
- Sum 46 appears once: \( P(46) = \frac{1}{6} \)
- Sum 47 appears twice: \( P(47) = \frac{2}{6} = \frac{1}{3} \)
- Sum 48 appears once: \( P(48) = \frac{1}{6} \)
- Sum 49 appears once: \( P(49) = \frac{1}{6} \)

Thus, the probability distribution is:
\[
P(\text{sum}) = \begin{cases}
\frac{1}{6} & \text{for sum } 45, 46, 48, 49 \\
\frac{1}{3} & \text{for sum } 47 \\
\end{cases}
\]

### Part (b): Draw a histogram.

Looking at the histograms in the provided image, you need to match the distribution from part (a). The correct histogram should show:
- A bar of height \( \frac{1}{6} \) at 45, 46, 48, and 49.
- A bar of height \( \frac{1}{3} \) at 47.

The correct option, based on the height of the bars, is **D**.

### Part (c): Find the probability that the sum is even.

From the list of sums:
- Even sums are 46 and 48.
- The probability of each of these sums is \( \frac{1}{6} \).

Thus, the probability that the sum is even is:
\[
P(\text{even}) = P(46) + P(48) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}.
\]

Do you want more details or have any questions?

### 5 Related Questions:
1. What is the total number of possible sums when drawing from a larger set of numbered slips?
2. How would the probability distribution change if the slips were replaced after being drawn?
3. Can you calculate the variance of the probability distribution in part (a)?
4. How does the shape of the histogram change if there are more slips in the hat?
5. What is the probability that the sum is odd?

### Tip:
When solving probability questions involving combinations, writing down all possible outcomes helps ensure accuracy.

Four slips of paper numbered 22, 23, 24, and 25 are in a hat. You draw a slip, note the result, and then draw a second slip and note the result (without replacing the first). Find the probability distribution for the sum of the two slips.

Explore the probability distribution for the sum of two slips drawn from a hat numbered 22, 23, 24, and 25. The solution includes steps for finding probabilities and drawing histograms. Suitable for high school students studying probability and statistics.

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