Question

The game requires $5 to play, once the player is admitted, he
or she has the opportunity to take a chance with luck and pick from
the bag. If the player receives a M&M, the player loses. If the
player wins a Reese’s Pieces candy, the player wins. If the player
wins they may roll a dice for a second turn, if the die rolls on a
even number, they may pick from the bag once again with no extra
charge, if the player rolls a odd, their turn is over.

The outcome of this game can be shown as the following: Paying
the $5 for entry fee gets the player into the game where they face
a 4/79 chance of winning and a 75/79 chance of losing for the first
phase. This phase is shown as the following:

300 M&M’s + 16 Reese’s Pieces=316 Total Counts

Winning: 16/316 which is 4/79 chance ( M&Ms are possible
choices out of total count)

Losing: 300/316 which is 75/79 chance (Reece’s Pieces are
possible choices out of total count)

4/79*1

If the player happens to win, they have a second chance to
continue by rolling a single die. If they player can land the die
on a even number, they may turn the knob of the machine once again,
if they land on a odd number the player ends their turn completely.
This following by rolling the die gives the player a ½ win or lose
chance. There are the same number of even and odd numbers on a 6
sided die.

1 2 3 4 5 6

Even numbers: 3/6 which is ½ chance of winning ( 2,4,6 )

Odd numbers: 3/6 which is ½ chance of losing ( 1,3,5 )

Questions:

* Is the game fair? Show the mathematical calculations for the
expected value of winning the game.

* If the game is not fair, how could you change the game to
make it fair?

* What are the probabilities for each outcome?

Answer #1

Ques 1 - The game is not fair, since the Probability of loss claimed is 75/79 but the actual probability of loss is 75/77, which is less than claimed. (for solution consider the photo)

Expected value of winning = P[win]*Prize of win + P[loss]*0 - entry fee = (2/77)*Reese + (75/77)*0 - $5 = (2/77)*Reese - $5

Ques 2 - To make it fair, the part where die is rolled should be removed. If the person wins, he gets Reese else loose. Then only the game will be fair

Ques 3 - Probability of win = 2/77

Probability of loose = 75/77

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