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

Relevance
  • 3 years ago
    Favorite 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?

    • Brianna3 years agoReport

      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!

  • 3 years ago

    0 = 4/52(x) + 48/52(-1)

    0 = 4/52(x) - 48/52

    48/52 = 4/52(x)

    48 = 4x

    x = 12

  • 3 years ago

    0=4/52(x)+48/52(-1) -> 0=4/52(x)-48/52 ->

    0=4-48x -> 48x=4 -> x= .0833

Still have questions? Get your answers by asking now.