Question

In: Statistics and Probability

Roll three (6-sided) dice. Let X denote the maximum of the values that appear. a. Find...

Roll three (6-sided) dice. Let X denote the maximum of the values that appear.

a. Find P(X=1).??

b. Find P(X=2).??

c. Find P(X=3).

d. Find P(X=4).??

e. Find P(X=5).?

f. Find P(X=6).

[Hint: It might be helpful to first find the values of P(X?x).]

Solutions

Expert Solution

When three dice are rolled possible number of outcomes are 6*6*6= 216.

(a)

Out of 216 outcomes, one outcome has all 1 so

P(X=1) = 1/216

(b)

The maximum 2 can be possible if at least one die shows 2 and rest dice shows 1.

That is outcome can have all 2's (in one way) or 2 2s and 1 1s (in three ways) or 1 2s and 2 1s (in three ways).

The possible number of outcomes with maximum 2: 1+3+3 = 7

Therefore

P(X=2) = 7/216

(c)

The maximum 3 can be possible if at least one die shows 3 and rest dice shows 1 or 2.

The possible number of outcomes are: 19

P(X=3) = 19/216

(d)

The maximum 4 can be possible if at least one die shows 4 and rest dice shows 1 or 2 or 3.

The possible number of outcomes are: 37

P(X=4) = 37/216

(e)

The maximum 5 can be possible if at least one die shows 5 and rest dice shows 1 or 2 or 3 or 4.

The possible number of outcomes are: 61

P(X=5) = 61/216

(f)

The maximum 6 can be possible if at least one die shows 6 and rest dice shows 1 or 2 or 3 or 4 or 5.

The possible number of outcomes are: 91

P(X=6) = 91/216


Related Solutions

Consider some 8-sided dice. Roll two of these dice. Let X be the minimum of the...
Consider some 8-sided dice. Roll two of these dice. Let X be the minimum of the two values that appear. Let Y denote the maximum.   a) Find the joint mass p_X,Y (x,y). b) Compute p_X│Y (x│y) in all cases. Express your final answer in terms of a table.
You roll two fair dice, and denote the number they show by X and Y. Let...
You roll two fair dice, and denote the number they show by X and Y. Let U = min{X, Y } and V = max{X, Y }. Write down the joint probability mass function of (U, V ) and compute ρ(U, V ) i.e the correlation coefficient of U and V
You roll two 6-sided dice numbered 1 through 6. Let A be the event that the...
You roll two 6-sided dice numbered 1 through 6. Let A be the event that the first die shows the number 3, let B be the event that the second die shows a 5, and let E be the event that the sum of the two numbers showing is even. Compute P(A)and P(B)and then compute P(AlB). What does this tell you about events A and B?Hint: Remember that the sample space has 36 outcomes! Compute P(ElA)and compute P(E). What does...
Exercise 7-12: Roll a pair of fair 3-sided dice. Let F denote the number of dots...
Exercise 7-12: Roll a pair of fair 3-sided dice. Let F denote the number of dots on the first die and let T denote the total number of dots. Determine the Cov(F,T).
2. Roll a pair of unbiased dice. Let X be the maximum of the two faces...
2. Roll a pair of unbiased dice. Let X be the maximum of the two faces and Y be the sum of the two faces. What is the joint density of X and Y ?
Suppose two fair dice are rolled. Let X denote the product of the values on the...
Suppose two fair dice are rolled. Let X denote the product of the values on the dice and Y denote minimum of the two dice. Find E[X] and E[Y] Find Var X and Var Y Let Z=XY. Find E[Z]. Find Cov(X,Y) and Corr(X,Y) Find E[X|Y=1] and E[Y|X=1]
1. Three six-sided dice are rolled. Let X be the sum of the dice. Determine the...
1. Three six-sided dice are rolled. Let X be the sum of the dice. Determine the range of X and compute P(X = 18) and P(X ≤ 4). 2. An urn contains 5 red balls and 3 green balls. (a) Draw 3 balls with replacement. Let X be the number of red balls drawn. Determine the range of X and compute P(X = 3) and P(X 6= 1). (b) Draw 3 balls without replacement. Let Y be the number of...
. 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?
Find the conditional​ probability, in a single roll of two fair​ 6-sided dice, that neither die...
Find the conditional​ probability, in a single roll of two fair​ 6-sided dice, that neither die is a three​,given that the sum is greater than 7.
Suppose you roll two 6 sided dice, letting X be the sum of the numbers shown...
Suppose you roll two 6 sided dice, letting X be the sum of the numbers shown on the dice and Y be the number of dice that show an odd number. a) Find the joint pmf of <X,Y> b) Find the marginals pmf's for both variables. c) Are X and Y independent?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT