No matter what two loaded dice we have, it cannot happen that each of the sums

$2,3,...,12$ comes up with the same probability.

Proof. Suppose that we have a pair of dice such that each of the $11$ possible sums

$2,3,...,12$ occurs with the same probability, $1/11.$ Let pi be the probability that

the first die shows i$,$ and qi that the second does. Then the probability that the sum

is $2$ is p$1$q$1,$ and the probability that it is $12$ is p$6$q$6.$ Consequently, p$1$q$1=1/11$

and p$6$q$6=1/11.$ Also, the probability that the sum is $7$ is at least p$1$q$6+$ p$6$q$1,$

so p$1$q$6+$ p$6$q$1<=1/11.$ Hence, q$1=1/11$p$1,$ q$6=1/11$p$6$ and

Answer rating: 100% (QA)

the probabilities of the first die showing ... View the full answer