Question

In: Math

Consider the following experiment: we roll a fair die twice. The two rolls are independent events....

Consider the following experiment: we roll a fair die twice. The two rolls are independent events. Let’s call M the number of dots in the first roll and N the number of dots in the second roll.

(a) What is the probability that both M and N are even?

(b) What is the probability that M + N is even?

(c) What is the probability that M + N = 5?

(d) We know that M + N = 5. What is the probability that M is an odd number?

(e) We know that M is an odd number. What is the probability that M + N = 5?

Expert Solution

(a) What is the probability that both M and N are even?
The total combinations on rolling a dice are given below.
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Favourable event set is { (2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6) } = 9 possiblilities.

Hence the needed probability = 9/36 = 1/4

(b) What is the probability that M + N is even?

The figure below shows all possible sums on rolling the two dice.

From these those highlighted in blue are favorable event set, which is 18 possibliities.

Hence the needed probability = 18/36 = 1/2

(c) What is the probability that M + N = 5?
The figure below shows all possible sums on rolling the two dice.

From these those highlighted in blue are favorable event set, which is 4 possibliities.

Hence the needed probability = 4/36 = 1/9

(d) We know that M + N = 5. What is the probability that M is an odd number?

P(A and B) = P(A given B) P(B)

Event A is M+N =5
Event B is M = odd number

P(A given B) = 1/9 (Found the previous problem)
P(B) = 3/6 = 1/2 ( 3 numbers are odd out of 6 numbers on dice)

P(A and B) = 1/9 * 1/2 = 1/18

(e) We know that M is an odd number. What is the probability that M + N = 5?

P(A and B) = P(A given B) P(B)
Event A is M = odd number
Event B is M+N =5

P(A given B) = 3/6 = 1/2 ( 3 numbers are odd out of 6 numbers on dice)
P(B) = 4/36

P(A and B) = 1/2 * 1/9 = 1/18

milcah answered 8 months ago

(1) For the random experiment “toss a fair coin twice and then roll a balanced die...

(1) For the random experiment “toss a fair coin twice and then roll a balanced die twice”, what’s the total count of the sample space? Let A be the event {both tosses show HEADs and both rolls show sixes}, B = {both tosses show the same side of the coin and both rolls show the same number}, C = {both tosses show the same side OR both rolls show the same number} and D = {the 1st roll of the...

Consider the following experiment: Simultaneously toss a fair coin and, independently, roll a fair die. Write...

Consider the following experiment: Simultaneously toss a fair coin and, independently, roll a fair die. Write out the 12 outcomes that comprise the sample space for this experiment. Let X be a random variable that takes the value of 1 if the coin shows “heads” and the value of 0 if the coin shows “tails”. Let Y be a random variable that takes the value of the “up” face of the tossed die. And let Z = X + Y....

A fair die is rolled twice. Let X be the maximum of the two rolls. Find...

A fair die is rolled twice. Let X be the maximum of the two rolls. Find the distribution of X. Let Y be the minimum of the two rolls. Find the variance of Y.

Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls...

Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls it takes to obtain the first 5, Y denote the number of rolls until the first 2, and Z denote the number of rolls until the first 4. Numerical answers are needed only for parts (a) and (b). Expressions are sufficient for parts (c), (d), and (e). a) E[X|Y = 1 or Z = 1] b) E[X|Y = 1 and Z = 2] c)...

The experiment of rolling a fair six-sided die twice and looking at the values of the...

The experiment of rolling a fair six-sided die twice and looking at the values of the faces that are facing up, has the following sample space. For example, the result (1,2) implies that the face that is up from the first die shows the value 1 and the value of the face that is up from the second die is 2. sample space of tossing 2 die A pair of dice is thrown. Let X = the number of multiples...

1. A fair coin is tossed twice. Consider the following events: a) A = a head...

1. A fair coin is tossed twice. Consider the following events: a) A = a head on the first toss; b) B = a tail on the first toss; c) C = a tail on the second toss; d) D = a head on the second toss. Are events A and B mutually exclusive ? WHY or WHY NOT ? Are events C and D Independent ? WHY or WHY NOT ? 2. Suppose there are 3 similar boxes. Box...

1. The experiment of rolling a fair six-sided die twice and looking at the values of...

1. The experiment of rolling a fair six-sided die twice and looking at the values of the faces that are facing up, has the following sample space. For example, the result (1,2) implies that the face that is up from the first die shows the value 1 and the value of the face that is up from the second die is 2. (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)...

You roll a balanced die two times. Is the second roll independent of the first roll?

Two rolls of a fair die. Let x and y be the results of the two...

Two rolls of a fair die. Let x and y be the results of the two rolls, and let z = x + y. (a) Find P[x = 4, y = 3], P[x > 3, y = 5]. (b) Find P[z = 7], P[z = 5], P[z = 3]. (c) Find the probability that at least one 6 appears given that z = 8. (d) Find the probability that x = 6 given that z > 8. (e) Find the...

Consider the following random variables based on a roll of a fair die, (1) You get...

Consider the following random variables based on a roll of a fair die, (1) You get $0.5 multiplied by the outcome of the die (2) You get $0 if the outcome is below or equal 3 and $1 otherwise (3) You get $1 if the outcome is below or equal 3 and $0 otherwise (4) You pay $1.5 if the outcome is below or equal 3 and get $1 multiplied the outcome otherwise Die Outcome Random Variable (1) Random Variable...