https://file.aiquickdraw.com/mathbot/file/1726543699213zrlwg2vl.jpg

The problem in the image asks to find the missing probability value \( K \) in a probability distribution table, and to compute the probability that a randomly selected individual donates blood more than 2 but less than 5 times a year.

### Part a: Finding the Missing Value \( K \)
The probability distribution must sum to 1, since it covers all possible outcomes. Therefore, we need to find the sum of the given probabilities and subtract from 1 to find \( K \).

Given probabilities:
\[
P(0) = 0.28, \quad P(1) = 0.22, \quad P(2) = 0.18, \quad P(3) = 0.14, \quad P(5) = 0.05, \quad P(6) = 0.02
\]

Summing these:
\[
0.28 + 0.22 + 0.18 + 0.14 + 0.05 + 0.02 = 0.89
\]

Now, the sum of all probabilities must equal 1, so:
\[
1 - 0.89 = 0.11
\]

Thus, \( K = 0.11 \).

### Part b: Probability of donating more than 2 but less than 5 times
This is the probability that \( X \) is 3 or 4. So we sum the probabilities for \( X = 3 \) and \( X = 4 \).

\[
P(3) + P(4) = 0.14 + 0.11 = 0.25
\]

Thus, the probability that a person donates more than 2 but less than 5 times a year is \( 0.25 \).

### Summary:
- \( K = 0.11 \)
- Probability of donating more than 2 but less than 5 times = 0.25

Let me know if you need further clarification or details!

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Here are some related questions:
1. What is the expected number of blood donations based on this distribution?
2. What is the variance of the number of donations?
3. How would the probability change if we wanted donations of 3 or more times?
4. What is the cumulative probability for donating 2 or fewer times?
5. How would you interpret the result of \( P(X > 6) \)?

**Tip:** In probability distributions, the sum of all possible outcomes must always equal 1.

A newspaper conducted a survey asking its readers how many times they donated blood each year. Let X represent the number of donations a randomly selected person who participated in the survey makes each year. The probability distribution of X is given as follows:

x: 0, 1, 2, 3, 4, 5, 6
P(x): 0.28, 0.22, 0.18, 0.14, K, 0.05, 0.02

Find the missing value K.
Find the probability that an individual selected at random from those surveyed donates blood more than 2, but less than 5 times a year.

Solve a probability distribution problem involving blood donation. Learn how to find the missing value K and calculate the probability of donating more than 2 but less than 5 times. Suitable for Grades 10-12, covering probability distribution concepts and the summation of probabilities.

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