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## Homework Statement

(see attachment)

## Homework Equations

## The Attempt at a Solution

The sum of the probabilities of each number on the dice is 1, i.e

[tex]\frac{1}{6}+\frac{1}{6}+\frac{1}{9}+x+\frac{2}{9}+y=1[/tex]

where x and y are the probabilities of number 4 and 6 respectively. Solving,

[tex]x+y=\frac{1}{3}[/tex]

The probability that the two dice shows same number is

[tex]\left(\frac{1}{6} \right)^2+\left(\frac{1}{6} \right)^2+\left(\frac{1}{9} \right)^2+x^2+\left(\frac{2}{9} \right)^2+y^2=\left(\frac{2}{3} \right)^4[/tex]

Solving, [tex]x^2+y^2=\frac{13}{18}[/tex]

Rewriting ##x^2+y^2## as ##(x+y)^2-2xy## and substituting the value of ##x+y##,

[tex]2xy=\frac{-11}{18}[/tex]

For the sum of two resulting numbers to be 10, there are three possible cases. The first shows 4 and the second shows 6 or (4,6). The other cases are (5,5) and (6,4).

The probability that the sum of the two resulting numbers is 10 can be given by the expression:

[tex]2xy+\left(\frac{2}{9} \right)^2[/tex]

Substituting the value of ##2xy##, I get a negative answer.

Any help is appreciated. Thanks!

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