Question

In: Statistics and Probability

You make a carnival game, where the player rolls two fair dice (in a single roll)...

You make a carnival game, where the player rolls two fair dice (in a single roll) and attempts to roll doubles (meaning both dice show the same number). The player puts down a dollar to play the game.

If the player loses, they lose their dollar.

If the player wins, they win $3 (and do not lose their original dollar). Answer the following (5 pts total).

  1. If you are running the game, what is the expected value of how much money you make running the game each time? Round to nearest cent.
  1. What is the standard deviation? Round to nearest cent.

  1. If you wanted to change the game to make it fair (so the expected value is zero), how could you change it?

Solutions

Expert Solution

Let X be the amount you payout to the player (You run the game). X = -$3 when the player wins (that is, roll a double and you payout $3) and X=$1 when the player loses (that is you make $1)

There are 6 ways to roll a double (11,22,33,...,66). Since the dice are fair, the probability of each being 1/6*1/6.

The probability of rolling a double is

That is, the probability that the player wins is 1/6

This is same as the probability that X=-3 is 1/6

The probability of player loses = the probability of not rolling a double = the probability of X=1 is =  1-1/6=5/6

We can get the following probability distribution of payout X

x P(x)
-3 0.1667
1 0.8333

a) If you are running the game, what is the expected value of how much money you make running the game each time? Round to nearest cent.

The expected value of X is

ans: If you are running the game, the expected value of how much money you make running the game each time is $0.33

b) What is the standard deviation? Round to nearest cent.

The expected value of is

the standard deviation of X is

ans: the standard deviation is $1.49

c) If you wanted to change the game to make it fair (so the expected value is zero), how could you change it?

Let us say you make the following changes in the payout

  • If the player loses, they lose their dollar.
  • If the player wins, they win $m (and do not lose their original dollar).

We need to find the value of m, so that the game is fair. A fair game has the expected value =0

The expected value of a fair game is

That is, if we payout $5 when the player wins, then the game would become fair.

ans: The modified game would be

The player puts down a dollar to play the game.

  • If the player loses, they lose their dollar.
  • If the player wins, they win $5 (and do not lose their original dollar).

orchestra answered 1 year ago
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