Question

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

• A. \(2.6\)
• B. \(3.8\)
• C. \(3.2\)
• D. \(3.6\)

Hint:

List the possible sums with their probabilities from the 10 equally likely pairs.

Answer 1

The Correct Option is D

Step 1: Linearity of expectation
The sum $X = X_1 + X_2$ for the two cards drawn. Each card drawn at random has the same expected value.

Step 2: Expected value of one card
$E(X_1) = \dfrac{1 + 1 + 2 + 2 + 3}{5} = \dfrac{9}{5} = 1.8$.

Step 3: Add
$E(X) = 2 \times 1.8 = 3.6$. This works for draws without replacement too, because each card is equally likely to be in either position.

Step 4: Verify
The long method with the distribution of $X$ also gives $\frac{36}{10} = 3.6$.

Final Answer:
The expected value is 3.6. This is option (D). \[ \boxed{\text{(D) }3.6} \]