Learn / Library / Expected value

$2$ cards without replacement

Expected valueHard

$6$ cards are numbered $1$ to $6$ and shuffled. $2$ are dealt without replacement, and the payout is the sum of their numbers. What is the variance of the payout?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

The $2$ cards are negatively correlated; the total of all $6$ is fixed.

Solution

One card has variance $σ^{2} = \frac{35}{12}$ .
The sum of all $6$ cards is constant, so $0 = 6 σ^{2} + 30 Cov$ , which gives $Cov = - \frac{σ ^{2}}{5} = - \frac{7}{12}$ for any $2$ cards.
The variance of the sum is $2 σ^{2} + 2 Cov = 2 \times \frac{35}{12} - 2 \times \frac{7}{12}$ .
That equals $\frac{14}{3}$ .

Answer: $\frac{14}{3}$

Next question Back to the library