# How To Solve This Equation?

To be honest, I have no clue how to get X by itself and solve for this... please help me. The question is:

You pick a card at random from an ordinary deck of 52 cards. If the card is an ace, you get 9 points; if not, you lose 1 point.

I somehow found out that the game is not fair because the expected value is -3/13.

What value for the aces would make the game fair? To find this, solve the equation:

0 = 4/52(x) + 48/52(−1)

I have virtually no idea how this equation works or what to start with. I need help! Thank you.

### 3 Answers

- PuzzlingLv 73 years agoFavorite Answer
Here's the logic.

If you draw an ace (probability = 4/52 = 1/13), you get 9 points.

If you draw any other card (probability = 48/52 = 12/13), you get -1 points.

If you multiply that out, you get the expected number of points per play:

E(X) = (1/13) * 9 + (12/13) * -1

= 9/13 - 12/13

= -3/13

So in the long run you are expected to lose with this game an average of 3/13 points per play.

If it is to be fair, the amount should be 0 (an average of nothing lost or won).

So instead of 9 points, let's use 'x' points. I'm also going to use 1/13 and 12/13 rather than the equivalent fractions of 4/52 and 48/52, but I think you see why they used that.

You want the expected winnings to be 0 in the long run to be fair.

(1/13)x + (12/13)(-1) = 0

x/13 - 12/13 = 0

Multiply everything by 13:

x - 12 = 0

x = 12

So you should be getting 12 points for drawing an ace for the game to be fair.

Got it?

I think I got it. I see how you got the answer, it just seems hard to understand over the internet. But I am starting to understand it better. Thank you!