Suppose you play a game of chance against a gambling house. In each round, you r

Question

Suppose you play a game of chance against a gambling house. In each round, you roll a pair of dice. If the sum comes up even, the house pays you that amount of money, in dollars. If the sum comes up odd, you pay the house that amount. For example, if you roll double sixes, then you pay the house $12, but if you roll 3-4, then the house pays you $7.

Is the game fair? If not, who is expected to win in the long run? (4 pts)

What is a 95% confidence interval for your winnings after 30 rounds? (6 pts)

Explanation / Answer

(The questions has a small error. In the question, it says that if the sum comes up even, the house is the one that pays but in the example if $12 which is even comes up, the house gets paid. The answer below follows the question, not example.)

The various sums possible and the payments are:

1 + 1 = 2 -> $2

1 + 2 = 3 -> -$3

1 + 3 = 4 -> $4

1 + 4 = 5 -> -$5

1 + 5 = 6 -> $6

1 + 6 = 7 -> -$7

2 + 1 = 2 -> -$3

2 + 2 = 4 -> $4

2 + 3 = 5 -> -$5

2 + 4 = 6 -> $6

2 + 5 = 7 -> -$7

2 + 6 = 8 -> $8

3 + 1 = 4 -> $4

3 + 2 = 5 -> -$5

3 + 3 = 6 -> $6

3 + 4 = 7 -> -$7

3 + 5 = 8 -> $8

3 + 6 = 9 -> -$9

4 + 1 = 5 -> -$5

4 + 2 = 6 -> $6

4 + 3 = 7 -> -$7

4 + 4 = 5 -> $8

4 + 5 = 9 -> -$9

4 + 6 = 10 -> $10

5 + 1 = 6 -> $6

5 + 2 = 7 -> -$7

5 + 3 = 8 -> $8

5 + 4 = 9 -> -$9

5 + 5 = 10 -> $10

5 + 6 = 11 -> -$11

6 + 1 = 7 -> -$7

6 + 2 = 8 -> $8

6 + 3 = 9 -> -$9

6 + 4 = 10 -> $10

6 + 5 = 11 -> -$11

6 + 6 = 12 -> $12

Note that for every single value of the first roll, the sum of the expected values is alternately -3 and 3. For e.g if the first roll comes up 1, the sum is $2 - $3 + $4 - $5 + $6 - $7.

Therefore, the overall expected value = 1/36 * (- $3 + $3 - $3 + $3 - $3 + $3) = 0.

Thus the game is fair.

Standard deviation = 1/36 (22 + 32 + 42 + 52 + 62 + 72 + 32 + 42 + 52 + 62 + 72 + 82 + 42 + 52 + 62 + 72 + 82 + 92 + 52 + 62 + 72 + 82 + 92 + 102 + 62 + 72 + 82 + 92 + 102 + 112 + 72 + 82 + 92 + 102 + 112 + 122) - 0 = 54.8333.

After 30 rounds, the sample mean is x' = 30 * 0 = 0.

The standard deviation is s = 54.8333 * 30 = 1645.

z for 95% = 1.96.

LCL = x' - z s / n = 0 - 1.96 * 1645 = -3224.2.

UCL = x' + z s / n = 0 + 1.96 * 1645 = 3224.2.

The interval is (-3224.2, 3224.2).

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