Question

There are two similar games played for Chinese New Year and Vietnamese New Year. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being \(1. The player places a bet on a number or object. The “house” rolls three dice. If none of the dice show the number or object that was bet, the house keeps the \)1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his or her \(1 bet, plus \)1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his or her \(1 bet, plus \)2 profit. If all three dice show the number or object bet, the player gets back his or her \(1 bet, plus \)3 profit. Let X = number of matches and Y = profit per game.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. List the values that Y may take on. Then, construct one PDF table that includes both X and Y and their probabilities.

e. Calculate the average expected matches over the long run of playing this game for the player.

f. Calculate the average expected earnings over the long run of playing this game for the player

g. Determine who has the advantage, the player or the house.

Short answer

a. X is the number of matches.

b. The values of X are $0,1,2,3$.

c. The distribution of X is $X~B(3,\frac{1}{6})$

d. The values of Y is $Y=(-1,1,2,3)$

e. The average expected matches over the long run of playing this game for the player is $\frac{1}{2}$

f. The average expected earnings over the long run of playing this game for the player is $-0.0787$

g. The house has the advantage.

Step-by-step solution

Content Introduction

The binomial distribution determines the probability of looking at a specific quantity of a successful results in a specific quantity of trials.

Explanation (part a)

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case the random variable X is the number of matches.

Explanation (part b)

Make the list of values that you want to use for X, so, we see there is an upper bound for the situation at hand $3$, then X is given by

$X=0,1,2,3$

Explanation (part c)

The probability distribution of binomial distribution has two parameters $n=numberoftrialsandp=probabilityofsuccess.$

The binomial distribution is of the form: $X~B(n,p)$

where $$\begin{gathered} n=3 \\ p=\frac{1}{6} \end{gathered}$$

Hence, the distribution is$X~B(3,\frac{1}{6})$

Explanation (part d)

Make the list of values that you want to use Y, so, we see there is an upper bound for the situation at hand $3$, then Y is given by:

$Y=-1,1,2,3$

Explanation (part e)

The average of expected matches over the long run of playing this game for the player is calculated using binomial distribution formula:$\mu =np$, where $\mu$is the average of expected matches over the long run of playing this game , $n=3$is number of trials and $p=\frac{1}{6}$is probability of success.

$$\begin{gathered} \mu =np \\ \mu =3\times \frac{1}{6} \\ \mu =\frac{1}{2} \end{gathered}$$

Explanation (part f)

the average expected earnings over the long run of playing this game for the player will be when we multiply the values of Y with probability of X.

Hence, the average expected earnings will be for the player playing this game is$-0.0787$

Explanation (part g)

The house has more advantage because the profit depends on the rolling of the dice.