Two cards are drawn at random from a box which contains 5 cards numbered 1, 1, 2, 2 and 3. If X denotes the sum of the numbers, then the expected sum is

Options

A. `3.75`
B. 4
C. `1.8`
D. 2

Answer

To find the expected sum \( E(X) \) when two cards are drawn from a box containing cards numbered 1, 1, 2, 2, and 3, we will follow these steps:

### Step 1: Identify all possible outcomes

The cards in the box are: 1, 1, 2, 2, 3. We will calculate the possible sums when drawing two cards. The possible pairs and their sums are:

1. (1, 1) → Sum = 2
2. (1, 2) → Sum = 3
3. (1, 2) → Sum = 3 (this is the same as the previous pair)
4. (1, 3) → Sum = 4
5. (2, 2) → Sum = 4
6. (2, 3) → Sum = 5

Thus, the unique sums from these pairs are: 2, 3, 4, and 5.

### Step 2: Calculate the probabilities of each sum

Next, we need to determine the probability of each sum occurring. We will count how many ways each sum can occur and divide by the total number of ways to choose 2 cards from 5.

The total number of ways to choose 2 cards from 5 is given by \( \binom{5}{2} = 10 \).

Now, we count the occurrences of each sum:

- **Sum = 2**: (1, 1) → 1 way
- **Sum = 3**: (1, 2) and (1, 2) → 2 ways (but counted as 1 unique way)
- **Sum = 4**: (1, 3) and (2, 2) → 3 ways
- **Sum = 5**: (2, 3) → 2 ways

Now, we can summarize the probabilities:

- \( P(X = 2) = \frac{1}{10} \)
- \( P(X = 3) = \frac{2}{10} = \frac{1}{5} \)
- \( P(X = 4) = \frac{3}{10} \)
- \( P(X = 5) = \frac{2}{10} = \frac{1}{5} \)

### Step 3: Calculate the expected value \( E(X) \)

The expected value \( E(X) \) is calculated using the formula:

\[ E(X) = \sum (x \cdot P(X = x)) \]

Calculating each term:

- For \( x = 2 \): \( 2 \cdot P(X = 2) = 2 \cdot \frac{1}{10} = \frac{2}{10} \)
- For \( x = 3 \): \( 3 \cdot P(X = 3) = 3 \cdot \frac{1}{5} = \frac{3}{5} = \frac{6}{10} \)
- For \( x = 4 \): \( 4 \cdot P(X = 4) = 4 \cdot \frac{3}{10} = \frac{12}{10} \)
- For \( x = 5 \): \( 5 \cdot P(X = 5) = 5 \cdot \frac{1}{5} = 1 = \frac{10}{10} \)

Now, summing these values:

\[ E(X) = \frac{2}{10} + \frac{6}{10} + \frac{12}{10} + \frac{10}{10} = \frac{30}{10} = 3 \]

### Final Answer

Thus, the expected sum \( E(X) \) is **3**.

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