Suppose we play a game where you pick a value between 0 and 19
(inclusive). A random value from 0 to 19 is chosen. If the values
match you win 6. If the values differ by at most one, you win 4. ( we
consider 0 and 19 to differ by one for this) Otherwise you lose 1.
what is your expected gain?

Hi nivis!
I will assume here that the random number is drawn from a uniform
distribution, and that only an integer number con be drawn or chosen
in the first place. So any integer between 0 and 19 has the same
probability of being chosen. If this assumption is not correct, please
request a clarification.

In order to find the expected gain of playing this game, let's
consider what can happen. Given that you choose any one number, there
will be only one number among the 20 (19 plus the zero) that will win
you 6. There are two number that will win you 4. And all the other
numbers (17 numbers) will make you loose 1.

For example, if you choose 14; then you would win 6 with a 14, you
would win 4 with either a 13 or 15, and you would win -1 with any
other number. This is the same with any number you can choose, even
for 19 or 0, because they are considered to differ by one.

Given that there are twenty numbers that can be randomly drawn, there
is one chance in twenty that you win 6, two in twenty that you win 4,
and 17 in twenty that you win -1. Therefore, the expected gain of this
game is:

(1/20)*6 + (2/20)*4 +(17/20)*(-1) = -0.15

That is, the expected LOSS from this game is 0.15. If you want to
learn more about expected value, please visit the following link

http://math.gmu.edu/~tlim/expect.htm


Google search strategy:
expected value
://www.google.com/search?sourceid=navclient&ie=UTF-8&oe=UTF-8&q=expected+value


I hope this helps! If there's anything unclear about my answer, please
don't hesitate to request a clarification before rating it. Otherwise,
I await your rating and final comments.


Best wishes!
elmarto

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Request for Answer Clarification by nivis-ga on 20 Sep 2003 20:05 PDT

Thanks I understood your answer. But one thing i want to clarify that
what is inclusive that in my question. Is that make any difference if
it exclusive.
Please reply me

nidhdo

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Clarification of Answer by elmarto-ga on 21 Sep 2003 13:04 PDT

Hi again nivis!
I'm glad you understood my answer. From your question, I understand
that inclusive means that 19 is a number that can be chosen and drawn
randomly; so there are 20 numbers to choose from: 1-19 and 0. If it
were "exclusive", the range of possible numbers would be from 0 to 18;
so there would be 19 numbers to choose from. This would change the
expected gain from this game. Instead of having one chance in twenty
to draw the number that wins you 6, you would have one in nineteen,
and the same for the other possibilities. In particular, the expected
gain would be:

(1/19)*6 + (2/19)*4 + (16/19)*(-1) = -0.105...

Best wishes!
elmarto