Question

A roulette wheel has 38 slots, numbered 1-36, 0 and 00. This means there are 18 even numbered slots, 18 odd numbered slots and two slots that are neither even nor odd. A ball is released into the wheel as it spins and is equally likely to land in any one of the slots. Find the amount each of the following gamblers is expected to win as well as the standard deviation.

(a) A gambler places a $100 bet on EVEN. If the ball lands in an even numbered pocket he wins $100, otherwise he loses $100. (It pays 1 to 1.)

(b) A second gambler decides to bet $100 on a single number instead since it pays 35 to 1. If the ball lands in his selected # slot, he wins $3500 dollars but he loses $100 if the ball lands in any other # slot.

Answer #1

a)

let x be the expected amount:

x | P(x) | xP(x) |
x^{2}P(x) |

100 | 9/19 | 47.368 | 4736.842 |

-100 | 10/19 | -52.632 | 5263.158 |

total | -5.263 | 10000.000 | |

E(x) =μ= | ΣxP(x) = | -5.2632 | |

E(x^{2}) = |
Σx^{2}P(x) = |
10000.0000 | |

Var(x)=σ^{2} = |
E(x^{2})-(E(x))^{2}= |
9972.299 | |

std deviation= |
σ= √σ^{2} = |
99.8614 |

from above expected win =-5.26

and std deviation=99.86

b)

x | P(x) | xP(x) |
x^{2}P(x) |

3500 | 1/38 | 92.105 | 322368.421 |

-100 | 37/38 | -97.368 | 9736.842 |

total | -5.263 | 332105.263 | |

E(x) =μ= | ΣxP(x) = | -5.2632 | |

E(x^{2}) = |
Σx^{2}P(x) = |
332105.2632 | |

Var(x)=σ^{2} = |
E(x^{2})-(E(x))^{2}= |
332077.562 | |

std deviation= |
σ= √σ^{2} = |
576.2617 |

expected win =-5.26

and std deviation=576.26

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