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: Count the outcomes
Two cards from five can be chosen in \({}^5C_2 = 10\) ways. The cards are \(1_a, 1_b, 2_a, 2_b, 3\).

Step 2: Distribution of X
Sum 2: the two 1s, 1 way. Sum 3: one 1 and one 2, \(2 \times 2 = 4\) ways. Sum 4: (1, 3) in 2 ways and (2, 2) in 1 way, total 3 ways. Sum 5: (2, 3) in 2 ways. Check: \(1 + 4 + 3 + 2 = 10\).

Step 3: Expected value
\[ E(X) = \frac{2(1) + 3(4) + 4(3) + 5(2)}{10} = \frac{2 + 12 + 12 + 10}{10} = \frac{36}{10} = 3.6 \]

Step 4: Check the options
\(3.8\) would come from counting the sum of 4 as 4 ways. \(2.6\) and \(3.2\) are too low for these sums. The value is 3.6, option (D).

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