The table below shows the probability distribution for X.
X Value	1	2	3	4	5
p(X)	1/12	2/12	2/12	3/12	4/12
Suppose you take a random sample of one score from this bucket. The probability that this score shows a 3 is p(X = 3) =0.42   . The probability that this score shows a number less than 3 is p(X < 3) =    . The probability that this score shows a number greater than 3 is p(X > 3) =    .

We are given the probability distribution for \( X \):

\[
\begin{array}{|c|c|}
\hline
X \text{ Value} & p(X) \\
\hline
1 & \frac{1}{12} \\
2 & \frac{2}{12} \\
3 & \frac{2}{12} \\
4 & \frac{3}{12} \\
5 & \frac{4}{12} \\
\hline
\end{array}
\]

Now, let's compute the probabilities based on the given distribution.

### 1. **Probability that the score shows a 3:**

This is directly given in the problem:

\[
p(X = 3) = \frac{2}{12} = 0.42
\]

### 2. **Probability that the score is less than 3:**

This means \( p(X < 3) \), which refers to the sum of probabilities for \( X = 1 \) and \( X = 2 \):

\[
p(X < 3) = p(X = 1) + p(X = 2) = \frac{1}{12} + \frac{2}{12} = \frac{3}{12} = 0.25
\]

### 3. **Probability that the score is greater than 3:**

This means \( p(X > 3) \), which refers to the sum of probabilities for \( X = 4 \) and \( X = 5 \):

\[
p(X > 3) = p(X = 4) + p(X = 5) = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} = 0.58
\]

### Summary:
- \( p(X = 3) = 0.42 \)
- \( p(X < 3) = 0.25 \)
- \( p(X > 3) = 0.58 \)

Let me know if you need further details or clarifications.

### Related Questions:
1. What is the probability that the score is exactly 4?
2. How would you calculate the expected value of \( X \) for this distribution?
3. What is the cumulative probability distribution for this table?
4. How can we calculate the variance and standard deviation for this probability distribution?
5. What happens to the probabilities if the values of \( X \) were shifted by a constant?

**Tip**: When computing probabilities involving inequalities like \( X < 3 \) or \( X > 3 \), simply sum the probabilities of the relevant values of \( X \).

The table below shows the probability distribution for X. X Value 1 2 3 4 5 p(X) 1/12 2/12 2/12 3/12 4/12 Suppose you take a random sample of one score from this bucket. The probability that this score shows a 3 is p(X = 3) = 0.42. The probability that this score shows a number less than 3 is p(X < 3) = . The probability that this score shows a number greater than 3 is p(X > 3) = .

Learn how to calculate probabilities from a probability distribution for the random variable X. This example shows how to determine the probability that X equals a specific value (P(X = 3)), and how to compute the probabilities for values less than and greater than 3. Suitable for students in Grades 8-10 studying basic probability.

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