At the school carnival. Ann is playing a game involving a stack of 10 index cards. Each card has a single number written on it: 1 card has a 1, 2 cards have a 2, 3 card have a 3, and 4 cards have a 4. Ann will choose 1 card at random, and she will be awarded the number of points equal to the number written on the card. Let the radom variable X represent the number of points Ann receives on any 1 draw. What is the expected value of X?

Options

A. 0.4
B. 1
C. 2.5
D. 3

Answer

To find the expected value of the random variable \( X \), which represents the number of points Ann receives from drawing a card, we can follow these steps:

### Step 1: Identify the outcomes and their probabilities
We have a stack of 10 index cards with the following distribution:
- 1 card has the number 1
- 2 cards have the number 2
- 3 cards have the number 3
- 4 cards have the number 4

We can summarize this information in a table:

| Points (X) | Number of Cards | Probability \( P(X) \) |
|------------|------------------|------------------------|
| 1 | 1 | \( \frac{1}{10} \) |
| 2 | 2 | \( \frac{2}{10} \) |
| 3 | 3 | \( \frac{3}{10} \) |
| 4 | 4 | \( \frac{4}{10} \) |

### Step 2: Verify that the probabilities sum to 1
To ensure that our probabilities are correct, we can add them up:
\[ \frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10} = \frac{1 + 2 + 3 + 4}{10} = \frac{10}{10} = 1 \]

### Step 3: Calculate the expected value \( E(X) \)
The expected value \( E(X) \) is calculated using the formula:
\[ E(X) = \sum (X \cdot P(X)) \]
Substituting the values we have:
\[ E(X) = 1 \cdot \frac{1}{10} + 2 \cdot \frac{2}{10} + 3 \cdot \frac{3}{10} + 4 \cdot \frac{4}{10} \]

Calculating each term:
- \( 1 \cdot \frac{1}{10} = \frac{1}{10} \)
- \( 2 \cdot \frac{2}{10} = \frac{4}{10} \)
- \( 3 \cdot \frac{3}{10} = \frac{9}{10} \)
- \( 4 \cdot \frac{4}{10} = \frac{16}{10} \)

Now, summing these values:
\[ E(X) = \frac{1}{10} + \frac{4}{10} + \frac{9}{10} + \frac{16}{10} = \frac{1 + 4 + 9 + 16}{10} = \frac{30}{10} = 3 \]

### Conclusion
The expected value \( E(X) \) is 3.

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