Two Dice Probability - Events X, Y and Z

[Little_Louis]

Question: Two fair sided dice with numbers 1 to 6 on the sides are thrown and the two numbers uppermost noted. The events X, Y and Z are defined as follows

X = the total of the two numbers is even

Y = the maximum of the two numbers is less than 5

Z = the total of the two numbers is greater than 9

Which one of the following statements is always correct?

1. The probability of X is less than that of Y
2. The probability of Z is 0.25
3. If Z occurs then the probability of X also occurring is greater than it would otherwise be
4. The probability that both X and Y occur is greater than 0.25

My Solution:

Plot the sample space for two dice.

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

X - the total number of evens in the sample space is 18 pairs = 18/36 = probability of 0.5
Y - the largest of the two numbers is less than 5 - so there are 16 pairs, so I think this is 16/36 = probability of 0.44 (2dp)
Z - the total of two numbers is greater than 9 - so there are pairs, 6/36 = probability 0.17

1. The probability of X is less than that of Y (0.5 > 0.44 (2dp) FALSE
2. The probability of Z is 0.25 (Z = 0.17) FALSE
3. If Z occurs then the probability of X also occurring is greater than it would otherwise be (Z + X/Z) = (0.25 + 0.5)/0.25 = 3 TRUE
4. The probability that both X and Y occur is greater than 0.25 (0.5 x 0.44 = 0.22) FALSE

I think out of the 4 statements, no 3 is the correct option. Please could someone confirm I have approached/completed this correctly. Many thanks.

 

[blamocur]

Question: Two fair sided dice with numbers 1 to 6 on the sides are thrown and the two numbers uppermost noted. The events X, Y and Z are defined as follows

X = the total of the two numbers is even

Y = the maximum of the two numbers is less than 5

Z = the total of the two numbers is greater than 9

Which one of the following statements is always correct?

1. The probability of X is less than that of Y
2. The probability of Z is 0.25
3. If Z occurs then the probability of X also occurring is greater than it would otherwise be
4. The probability that both X and Y occur is greater than 0.25

My Solution:

Plot the sample space for two dice.

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

X - the total number of evens in the sample space is 18 pairs = 18/36 = probability of 0.5
Y - the largest of the two numbers is less than 5 - so there are 16 pairs, so I think this is 16/36 = probability of 0.44 (2dp)
Z - the total of two numbers is greater than 9 - so there are pairs, 6/36 = probability 0.17

1. The probability of X is less than that of Y (0.5 > 0.44 (2dp) FALSE
2. The probability of Z is 0.25 (Z = 0.17) FALSE
3. If Z occurs then the probability of X also occurring is greater than it would otherwise be (Z + X/Z) = (0.25 + 0.5)/0.25 = 3 TRUE
4. The probability that both X and Y occur is greater than 0.25 (0.5 x 0.44 = 0.22) FALSE

I think out of the 4 statements, no 3 is the correct option. Please could someone confirm I have approached/completed this correctly. Many thanks.

The final answer looks correct to me, but:
In 3.: I don't understand your reasoning. I.e., what does this mean: (Z + X/Z) = (0.25 + 0.5)/0.25 = 3
In 4.: you probably want to prove, or at least mention, that X and Z are independent. Otherwise you need to show that multiplying probability is applicable.

 

[Little_Louis]

The final answer looks correct to me, but:
In 3.: I don't understand your reasoning. I.e., what does this mean: (Z + X/Z) = (0.25 + 0.5)/0.25 = 3
In 4.: you probably want to prove, or at least mention, that X and Z are independent. Otherwise you need to show that multiplying probability is applicable.

@blamocur thanks for the response!
In 3, I have worked out the probability of X given that Z has occurred P(X|Z) = P(Z + X)/P(Z)
In 4 P(X ∩ Y) = P(X) * P(Y)
Is that better?

 

[pka]

Question: Two fair sided dice with numbers 1 to 6 on the sides are thrown and the two numbers uppermost noted. The events X, Y and Z are defined as follows
X = the total of the two numbers is even
Y = the maximum of the two numbers is less than 5
Z = the total of the two numbers is greater than 9
Which one of the following statements is always correct?The probability of X is less than that of Y

1. The probability of Z is 0.25
2. If Z occurs then the probability of X also occurring is greater than it would otherwise be
3. The probability that both X and Y occur is greater than 0.25

My approach is to use a generating polynomial:
[imath]{\left( {\sum\limits_{k = 1}^6 {{x^k}} } \right)^2}=x^{12} + 2 x^{11} + 3 x^{10} + 4 x^9 + 5 x^8 + 6 x^7 + 5 x^6 + 4 x^5 + 3 x^4 + 2 x^3 + x^2[/imath]
In this expansion each term answers a question. The sum of the coefficients is [imath]36[/imath].
The term [imath]3x^{10}[/imath] tells us that there are three ways to toss a sum of ten.
Thus [imath]\mathscr{P}(X)=\dfrac{18}{36}[/imath]

 

[blamocur]

@blamocur thanks for the response!
In 3, I have worked out the probability of X given that Z has occurred P(X|Z) = P(Z + X)/P(Z)
In 4 P(X ∩ Y) = P(X) * P(Y)
Is that better?

Not really: I don't see how this addresses my suggestions. But, as I said, the answers are correct, and it is possible that your teacher does not expect any more than that.