Question

A toy roulette wheel has 9 slots (not 38 slots like the real thing) which are numbered 0, 1, 2, ..., 8. Jack and Jill play the following game 180 times in a row: Jack will win 3 pennies from Jill if a 4 or an 8 comes up; otherwise he will lose 1 penny.

After 180 games we expect Jack to have

		won a total of 0 pennies
		won a total of 0.11 pennies
		won a total of 9 pennies
		won a total of 20 pennies
		lost a total of 0 pennies
		lost a total of 0.11 pennies

Answer 1

Define J= winning amount of Jack in a single game.

X= number appeared on the wheel.

Then J= 3 if X=4 or 8 ,

   J=-1 otherwise.

We assume that each number appears with equal probability , that is P(X=i)=1/9,i=0,1,..,8.

Then E(J)=3*P(X=4 or 8)-1*(1-P(X=4 or 8))

=3*2/9-1*7/9=-1/9

Thus in 180 games, expected gain of Jack=-180/9=-20 pennies. or Jack will lost a total of 20 pennies on average.

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