Question 4: Chuck-a-Luck There is a game called Chuck-a-Luck. You pick a number

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Question 4: Chuck-a-Luck There is a game called Chuck-a-Luck. You pick a number from 1 to 6. You roll three dice. . If your number doesn't appear on any dice, you lose $1. . If your number appears exactly once, you win SI If your number appears on exactly two dice, you win $2. . If your number appears on all three dice, you win $3. 1. Complete the following probability distribution table, where the random variable is the number of times your lucky number appears out of three dice. (Hint: Use Binomial distribution) Chuck-A-Luck Probability Distribution Times your lucky number appears Probability 2. Complete the following probability distribution table, where the random variable is the monetary outcome of the game (dollars you gain or lose). Chuck-A-Luck Probability Distribution Dollars you gain from the game Probability 3. Should you be willing to play this game? (Hint: Calculate the expected value. If the expected value of your gain >0, you are on average going to make money. If the expected on average going to lose money) valuec0, you are

Explanation / Answer

1.

Probability that chosen number appears any of the dice = 1/6

Probability that chosen number does not appears any of the dice = 5/6

Using binomial distribution, where the number of trials, n = 3 and the probability of success, P = 1/6. The probability distribution of x (Times your lucky number appears) is given as,

P(X = x) = nCx * Px * (1 - P)n-x

P(X = x) = 3Cx * (1/6)x * (5/6)3-x

Putting x = 0, 1, 2, 3, we get

2.

The monetary outcome of the game is -1, 1, 2, 3 when the Times your lucky number appears is 0, 1, 2, 3 respectively.

So, the distribution of Times your lucky number appears is same as Dollars you gain from the game

3.

Expected value of game = Sum of product of probabilities with Dollars you gain from the game

= 0.5787 * (-1) + 0.3472 * 1 + 0.0695 * 2 + 0.0046 * 3

= -0.0787

As, the expected value of the game < 0, we are on average going to lose money. So, we will not be willing to play this game.

Times your lucky number appears Probability 0 0.5787 1 0.3472 2 0.0695 3 0.0046

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