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Math Help - expected value

  1. #1
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    expected value

    There are ten evolopes in a box. Each envolope contains a single bill. Of the bills, five are one-dollar bills, two are five-dollar bills, two are ten-dollar bills and one is a twenty dollar bill.

    a) Find the expected value of the draw. Interpret the meaning of this expected value.

    b) If you were the casino operator of this game, what would you chared people to select an envolope? Explain. Refer to answer in part a. The cosino wants to make money to keep the customers playing.


    Help me understand. Thanks =)
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  2. #2
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    Quote Originally Posted by l0v3n View Post
    There are ten evolopes in a box. Each envolope contains a single bill. Of the bills, five are one-dollar bills, two are five-dollar bills, two are ten-dollar bills and one is a twenty dollar bill.

    a) Find the expected value of the draw. Interpret the meaning of this expected value.

    b) If you were the casino operator of this game, what would you chared people to select an envolope? Explain. Refer to answer in part a. The cosino wants to make money to keep the customers playing.


    Help me understand. Thanks =)
    a) Let X be the random variable 'amount of money ($) in drawn envelope'.

    E(X) = (5/10)(1) + (2/10)(5) + (2/10)(10) + (1/10)(20) = 5.50.

    b) Charge more than $5.50.
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  3. #3
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    Hello, l0v3n!

    There are ten evelopes in a box. Each envelope contains a single bill.
    There are five $1-bills, two $5-bills, two $10-dollar bills and one $20-bill.

    a) Find the expected value of the draw. Interpret the meaning of this expected value.

    . . \begin{array}{ccc}Pr(\$1) & = & \frac{5}{10} \\ \\[-2mm]<br />
Pr(\$5) &=& \frac{2}{10} \\ \\[-2mm]<br />
Pr(\$10) &=& \frac{2}{10} \\ \\[-2mm]<br />
Pr(\$20) &=&\frac{1}{10} \end{array}


    EV \;=\;(\$1)\left(\frac{5}{10}\right) + (\$5)\left(\frac{2}{10}\right) + (\$10)\left(\frac{2}{10}\right) + (\$20)\left(\frac{1}{10}\right) \;=\;\$5.50

    If you play this game repeatedly,
    . . you can expect to win an average of $5.50 per game.




    b) If you were the casino operator of this game,
    what would you charge people to select an envelope? Explain.
    Of course, we would charge more than $5.50 . . .
    . . otherwise we'd be giving money away.


    If we charge too much, no one will play the game.

    For example, it we charge $10 to play the game,
    . . they will win $20 only one-tenth of the time.
    The rest of the time they will lose $10 or break even.
    Their expected value is: . (-\$10)\left(\frac{7}{10}\right) + (\$0)\left(\frac{2}{10}\right) + (\$20)\left(\frac{1}{10}\right) \:=\:-\$5
    . . They can expect to lose an average of $5 per game.

    I would charge $6 per play.
    I can expect to make an average 50 cents over 10 plays, a nickel per play.
    And if the game were played 24 hours a day, 7 days a week . . .

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