Question

# A roulette wheel has 38 slots, numbered 1-36, 0 and 00. This means there are 18...

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.

a)

let x be the expected amount:

 x P(x) xP(x) x2P(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(x2) = Σx2P(x) = 10000.0000 Var(x)=σ2 = E(x2)-(E(x))2= 9972.299 std deviation= σ= √σ2 = 99.8614

from above expected win =-5.26

and std deviation=99.86

b)

 x P(x) xP(x) x2P(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(x2) = Σx2P(x) = 332105.2632 Var(x)=σ2 = E(x2)-(E(x))2= 332077.562 std deviation= σ= √σ2 = 576.2617

expected win =-5.26

and std deviation=576.26

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