In: Math

Consider the following game: Three cards are labeled $1, $4, and $7. A player pays a $9 entry fee, selects 2 cards at random without replacement, and then receives the sum of the winnings indicated on the 2 cards.

a) Calculate the expected value and standard deviation of the random variable "net winnings" (that is, winnings minus a $9 entry fee)

b) Suppose a 4th card, labelled *k,* is added to the game
but the player still selects two cards without replacement. What is
the value of *k* which makes the game fair (i.e makes
expected net winnings = $0)

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a).

P (X1) | 1/3 |

P (X2) | 1/2 |

X1 | X2 | Y (X1 + X2) |

1 | 4 | 5 |

1 | 7 | 8 |

4 | 1 | 5 |

7 | 1 | 8 |

4 | 7 | 11 |

7 | 4 | 11 |

therefore P (Y) = P (X1) * P (X2)

Y | Freq. | P (Y) | W (Y - 9) | P (W) | W * P (W) | W ^2 | W ^2 * P (W) |

5 | 2 | 1/3 | -4 | 1/3 | -1 1/3 | 16 | 5 1/3 |

8 | 2 | 1/3 | -1 | 1/3 | - 1/3 | 1 | 1/3 |

11 | 2 | 1/3 | 2 | 1/3 | 2/3 | 4 | 1 1/3 |

sum | 1 | 1 | -1 | 7 |

b ).

X1' | X2' | Y' (X1' +X2') |

1 | 4 | 5 |

1 | 7 | 8 |

1 | K | K + 1 |

4 | K | K + 4 |

4 | 1 | 5 |

4 | 7 | 11 |

7 | 4 | 11 |

7 | K | K + 7 |

7 | 1 | 8 |

K | 1 | K + 1 |

K | 7 | K + 7 |

K | 4 | K + 4 |

72 + 6K | ||

P (X1') | 1/4 | |

P (X2') | 1/3 |

Y' | Freq. | P (Y') | W (Y'- 9) |

11 | 2 | 1/6 | 2 |

5 | 2 | 1/6 | -4 |

8 | 2 | 1/6 | -1 |

K + 1 | 2 | 1/6 | K - 8 |

K + 4 | 2 | 1/6 | K - 5 |

K + 7 | 2 | 1/6 | K - 2 |

Sum | 12 | 1 | |

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