Question

In: Statistics and Probability

1)A casino wants to introduce a new game. In this game a player rolls two 4-sided...

  1. 1)A casino wants to introduce a new game. In this game a player rolls two 4-sided dice and their winnings are determined by the following rules:
    1. Otherwise, the player wins nothing.
    2. If 3 ≤ sum ≤ 5 the player wins $25.
    3. If sum ≥ 6 the player wins $100.
  1. a)Determine the probability distribution for this gamble.

Probability Distribution

X

P (X)

b)How much should the casino charge for this game if they want to make a profit in the long run? Show the calculations that support your decision

c)What is the standard error, σX, for this gamble

  1. Complete the probability distribution for X below.

Distribution of X

X

P (X = k)

P (X ≤ k)

0

1

0.4219

2

3

0.0469

4

0.0039
  1. Compute: (i) P (X ≤ 3) and (ii) P (X < 3)
  1. Compute: P (X > 1)
  1. Compute: P (X is odd)
  1. Compute the expected value and variance of X.

Solutions

Expert Solution

We would be looking at the first question all parts here:

a) For 2 four sided dice, the total number of combinations is computed as:
= 4*4 = 16

Therefore the probability distribution of the sum of the two numbers is obtained here as:

P(Sum = 2) = P(11) = 1/16
P(Sum = 3) = P(12, 21) = 2/16
P(Sum = 4) = P(13, 31, 22) = 3/16
P(Sum = 5) = P(14,23,42,41) = 4/16
P(Sum = 6) = P(24,42,33) = 3/16
P(Sum = 7) = P(34,43) = 2/16
P(Sum = 8) = 1/16

Using this, the PDF for the gamble is obtained here as:

P(X = 25) = P(3 <= Sum <= 5) = ( 2 + 3 + 4)/16 = 0.5625
P(X = 100) = P(Sum >= 6) = (3 + 2 + 1)/16 = 0.375
P(X = 0) = 1 - 0.5625 - 0.375 = 0.0625

Therefore the PDF for gamble here is given here as:
P(X = 0) = 0.0625,
P(X = 25) = 0.5625,
P(X = 100) = 0.375

b) To be profitable in long run, the casino should charge more than the expected value of the gamble to the player here. This is computed as:

Therefore the casino should charge more than 51.5625 here to be profitable in long run.

c) The second moment of X is first computed here as:

The standard deviation of X now is computed here as:

Therefore 37.9851 is the required standard deviation here.

orchestra answered 2 years ago
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