Q.3: Rolling two fair dice
a. Sample space
Each die has 6 faces and we have 2 die
There are a total of 6*6 = 36 possible outcomes.
Sample space = { (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) }
Part b:
X₁ is the face value of the 1st die
X₂ is the face value of the 2nd die
Y is the sum of two dice
(i) Pr (X₁ ≥ 5 ∩ X₂ = 2)
The probability that X₁ is equal to or greater than 5 is given by
There are 2 outcomes of interest X₁ = 5 and X₁ = 6 and the total number of outcomes are 6.

So, Pr (X₁ ≥ 5) = 1/3
The probability that X₂ is equal 2 is given by
There is only 1 outcome of interest that is X₂ = 2 and the total number of outcomes are 6.

So, Pr (X₂ = 2) = 1/6
The intersection of these two probabilities is given by

Therefore, Pr (X₁ ≥ 5 ∩ X₂ = 2) = 1/18
(ii) Pr (X₁ = 3 U X₂ = 3)
The probability that X₁ is equal 3 is given by
There is only 1 outcome of interest that is X₁ = 2 and the total number of outcomes are 6.

So, Pr (X₁ = 3) = 1/6
The probability that X₂ is equal 3 is also the same.
So, Pr (X₂ = 3) = 1/6
The union of these two probabilities is given by

Where P(X₁ = 3 ∩ X₂ = 3) is the intersection of the two probabilities.
This is the event when there is a 3 on both of the dice.
P(X₁ = 3 ∩ X₂ = 3) = 1/36

Therefore, Pr (X₁ = 3 U X₂ = 3) = 11/36
(iii) Pr (X₁ ≥ 3 | Y = 8)
This is a conditional probability which is given by

Let us find each of these probabilities
The probability that X₁ is equal to or greater than 3 is given by
There are 4 outcomes of interest X₁ = 3, X₁ = 4, X₁ = 5, and X₁ = 6 and the total number of outcomes are 6.

So, Pr (X₁ ≥ 3) = 2/3
The probability that Y = 8 is given by
Count the number of outcomes that will give us Y = 8
(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
So, there are 5 outcomes of interest and the total number of outcomes are 36.

So, Pr (Y = 8) = 5/36
The intersection of these two probabilities is given by

So, the conditional probability is

Therefore, Pr (X₁ ≥ 3 | Y = 8) = 2/3
(iv) Pr (X₂ < 5 | Y > 7)
This is a conditional probability which is given by

The probability that X₂ is less than 5 is given by
There are 4 outcomes of interest X₂ = 4, X₂ = 3, X₂ = 2, and X₂ = 1 and the total number of outcomes are 6.

The probability that Y > 7 is given by
Count the number of outcomes that will give us Y > 7
So, there are 16 outcomes of interest and the total number of outcomes are 36.

So, Pr (Y > 7) = 4/9
The intersection of these two probabilities is given by

So, the conditional probability is

Therefore, Pr (X₂ < 5 | Y > 7) = 2/3