Question

In: Statistics and Probability

Question: You have five dice, like in a game of Yahtzee! Suppose you roll the five...

Question:

You have five dice, like in a game of Yahtzee! Suppose you roll the five dice once and sum the numbers the five dice show.

(a) What is the mean and the standard deviation of the sum of five dice?

(b) Suppose you average 60 of such rolls with the five dice. What is the distribution of this average?

(c) What is the chance the average of 60 such rolls is larger than 18?

Expert Solution

When we roll a fair die possible outcome are 1, 2, 3, 4, 5, and 6. Since each outcome is equally likely so the probability of each outcome is 1/6 .

Following table shows the calculations:

Xi	P(Xi=xi)	Xi*P(X1=xi)	Xi^2*P(Xi=xi)
1	1/6	0.166666667	0.166666667
2	1/6	0.333333333	0.666666667
3	1/6	0.5	1.5
4	1/6	0.666666667	2.666666667
5	1/6	0.833333333	4.166666667
6	1/6	1	6
Total		3.5	15.16666667

So,

and

We roll five dice and let outcome of five dice are shown by X1, X2, X3, X4 and X5.

Let X= X1+X2+X3+X4+X5

The mean of the sum of five dice is

The standard deviation of the sum of five dice is

(b)

(c)

orchestra answered 1 year ago

1. A round in the game Yahtzee begins by rolling five fair dice. Find the probability...

1. A round in the game Yahtzee begins by rolling five fair dice. Find the probability of rolling a: a. one pair (ex 33421 but not 33441), two pair (ex 33441 but not 33444), and three of a kind (ex 24252 but not 24242) 2. Consider a 10x10 matrix that consists of all zeros. ten elements of the matrix are selected at random and their value is changed from a zero to a one. find the probability that the ones...

In Yahtzee, a “large straight” is a roll that leads to five dice displaying 1-2-3-4-5 or...

In Yahtzee, a “large straight” is a roll that leads to five dice displaying 1-2-3-4-5 or 2-3-4-5-6. What is the probability that a player will score a large straight with a single roll of all six dice?

for monopoly board game, Suppose you are on “GO”. On your next roll of the dice,...

for monopoly board game, Suppose you are on “GO”. On your next roll of the dice, what is the probability that: a. You land on Jail b. You land on any Light Blue property (Connecticut Ave, Vermont Ave, or Oriental Ave.) c. You land on Mediterranian Ave d. You land on Chance or Community Chest 2. Suppose you are on “Jail” a. What space (or spaces) is/are most likely to be landed on on your next roll? Justify your response....

Suppose you’re playing a game where you consecutively roll 3 dice. After each roll you may...

Suppose you’re playing a game where you consecutively roll 3 dice. After each roll you may choose to either roll the next dice or sacrifice one die to reroll any number of the previous dice. If you get a number greater than 5 you win, but if you roll doubles or a number less than 6 you lose. Considering each roll a separate state what is the approximate branching factor? Justify. Draw the full state space considering only the number...

This problem concerns the dice game craps. On the first roll of two dice, you win...

This problem concerns the dice game craps. On the first roll of two dice, you win instantly with a sum of 7 or 11 and lose instantly with a roll of 2,3, or 12. If you roll another sum, say 5, then you continue to roll until you either roll a 5 again (win) or roll a 7 (lose). How do you solve for the probability of winning?

suppose you roll two fair dice. A) what is the probability that you will roll an...

suppose you roll two fair dice. A) what is the probability that you will roll an even number on the first die AND a 5 on the second die B) What is the probability that the sum of the numbers on the two dice is 9? show all work.

We will simulate a dice game in which 2 dice are thrown. If the roll is...

We will simulate a dice game in which 2 dice are thrown. If the roll is 7 or 11, you win. If the roll is 2, 3, or 12, you lose If the roll is any other value, it establishes a point. If with a point established, that point is rolled again before a 7, you win. If, with a point established, a 7 is rolled before the point is rolled again you lose. Build your algorithm incrementally. First write...

In the game of Craps, you roll two dice. When you bet on a “snake eyes”,...

In the game of Craps, you roll two dice. When you bet on a “snake eyes”, meaning a 1 on both dice, you win $30 for each $1 you bet. Otherwise, you lose your dollar. What is the probability of winning this bet? What is the expected value of making this bet? If you play this game 100 times, how much would you expect to lose?

Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these...

Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers. (b) List all possible samples and calculate the value of the sample mean ¯(X) and variance (s 2 ) for each sample? (c) Obtain the sampling distribution of X¯ from this list by creating a frequency distribution table. You can create a frequency distribution...

. Three Dice of a Kind Consider the following game: You roll six 6-sided dice d1,…,d6...

. Three Dice of a Kind Consider the following game: You roll six 6-sided dice d1,…,d6 and you win if some number appears 3 or more times. For example, if you roll: (3,3,5,4,6,6) then you lose. If you roll (4,1,3,6,4,4) then you win. What is the probability that you win this game?