# A person throws two fair dice. He wins Rs. 15 for throwing a doublet ( same numbers on the two dice), wins Rs. 12 when the throw resultsin the sum of 9 , and loses Rs. 6 for any other outcome on the throw. Then the expected gain / loss (in Rs.) of the person is : Option 1) $\frac{1}{2}$  gain  Option 2) $\frac{1}{4}$ loss Option 3) $\frac{1}{2}$ loss Option 4) 2 gain

When 2 dice are thrown , Sample space would be

$\begin{Bmatrix} (1,1) &(2,1) ........... &(6,1) \\ (1,2) & (2,2) ...........&(6,2) \\ (1,3)& (2,3) ........... & (6,3)\\ (1,4)& .................. &(6,4) \\ (1,5)& ................... &(6,5) \\ (1,6) & ................... & (6,6) \end{Bmatrix}$

Now, the expected gain or loss = both + Sum (9) - other

$=(\frac{6}{36}\times 15)+(\frac{4}{36}\times 12)-(\frac{26}{36}\times 6)=-\frac{1}{2}$

So, there is loss of $Rs \: \: \frac{1}{2}$

Option 1)

$\frac{1}{2}$  gain

Option 2)

$\frac{1}{4}$ loss

Option 3)

$\frac{1}{2}$ loss

Option 4)

2 gain

Prob. of getting a doublet =1/6

Prob. of getting a sum of 9=1/9

Prob. of getting other outcomes=13/18

Now according to the ques.

After getting a doublet money gained by him=15×1/6 =5/2

After getting a sum of 9 money gained by him=1/9×12 =4/3

When there is any other outcome,he looses money=13/18×6 =13/3

So the complete loss or  gain

=5/2 + 4/3 - 13/3= -1/2

This implies his expected result is loss of money and he looses 1/2rs .

Option 1

Option 2) 1/4 loss

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