In a dice game, a player pays a stake of Re

    \[1\]

for each throw of a die. She receives Rs

    \[5\]

if the die shows a

    \[3\]

, Rs

    \[2\]

if the die shows a

    \[1\]

or

    \[6\]

, and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?
In a dice game, a player pays a stake of Re

    \[1\]

for each throw of a die. She receives Rs

    \[5\]

if the die shows a

    \[3\]

, Rs

    \[2\]

if the die shows a

    \[1\]

or

    \[6\]

, and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?

Let’s take X to be the random variable of profit per throw.

As, she loses Rs

    \[1\]

for giving any od

    \[2,4,5\]

.

So,

    \[P\left( X\text{ }=\text{ }-1 \right)\text{ }=\text{ }1/6\text{ }+\text{ }1/6\text{ }+\text{ }1/6\text{ }=\text{ }3/6\text{ }=\text{ 1/2}\]

    \[P\left( X\text{ }=\text{ }1 \right)\text{ }=\text{ }1/6\text{ }+\text{ }1/6\text{ }=\text{ }2/6\text{ }=\text{ }1/3\]

[Since, die showing

    \[1\]

or

    \[6\]

]

    \[P\left( X\text{ }=\text{ }4 \right)\text{ }=\text{ }1/6\]

[Since, die shows only a

    \[3\]

]

Thus, the player’s expected profit =

    \[\sum{{{p}_{1}}{{x}_{i}}}\]

    \[=\text{ }-1\text{ }\times \text{ 1/2 }+\text{ }1\text{ }\times \text{ }1/3\text{ }+\text{ }4\text{ }\times 1/6\text{ }=\text{ }-1/2\text{ }+\text{ }1/3\text{ }+\text{ }2/3\text{ }=\text{ 1/2 }=\text{ }Rs\text{ }0.50\]