A fair die is tossed once. If the random variable is the number of "getting an even number" (denoted by X), find the probability distribution of 'X'. Sketch the graph.

Answer

To find the probability distribution of the random variable \( X \), which represents the number of times an even number is rolled when a fair die is tossed once, we will follow these steps:

### Step 1: Identify the Sample Space
The sample space \( S \) when tossing a fair die consists of the outcomes:
\[ S = \{1, 2, 3, 4, 5, 6\} \]

### Step 2: Identify the Even and Odd Numbers
From the sample space, we can identify the even and odd numbers:
- Even numbers: \( \{2, 4, 6\} \)
- Odd numbers: \( \{1, 3, 5\} \)

### Step 3: Define the Random Variable \( X \)
The random variable \( X \) can take the following values:
- \( X = 0 \): if an odd number is rolled (1, 3, or 5)
- \( X = 1 \): if an even number is rolled (2, 4, or 6)

### Step 4: Calculate the Probabilities
Now, we will calculate the probabilities for each value of \( X \):
- Probability of \( X = 0 \) (rolling an odd number):
\[ P(X = 0) = \frac{\text{Number of odd outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]

- Probability of \( X = 1 \) (rolling an even number):
\[ P(X = 1) = \frac{\text{Number of even outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]

### Step 5: Write the Probability Distribution
The probability distribution of \( X \) can be summarized as follows:
\[ \begin{align*} X & : 0 \quad 1 \\ P(X) & : \frac{1}{2} \quad \frac{1}{2} \end{align*} \]

### Step 6: Sketch the Graph
To sketch the graph of the probability distribution:
- On the x-axis, plot the values of \( X \) (0 and 1).
- On the y-axis, plot the corresponding probabilities \( P(X) \).

The graph will have two bars:
- A bar at \( X = 0 \) with height \( \frac{1}{2} \)
- A bar at \( X = 1 \) with height \( \frac{1}{2} \)

### Final Summary
The probability distribution of \( X \) is:
- \( P(X = 0) = \frac{1}{2} \)
- \( P(X = 1) = \frac{1}{2} \)