Two integers x and y are chosen with replacement from the set 0,1,2,,10.

Question

Two integers $x$ and $y$ are chosen with replacement from the set $\{0,1,2,\dots,10\}$. Then the probability that $|x-y|>5$ is:

Accepted Answer

Total pairs $=11^2=121$. Counting pairs with $|x-y|>5$, i.e. $|x-y|\ge6$: for each difference $d=6,\dots,10$, the number of ordered pairs is $2(11-d)$.

$$\sum_{d=6}^{10} 2(11-d) = 2(5+4+3+2+1)=30$$

$$P = \frac{30}{121}$$

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