Simultaneous Throw of a Pair of Dice

A doublet:
A doublet is obtained when both dice show the same number. The probability of getting a doublet can be calculated by dividing the number of favorable outcomes (6) by the total number of possible outcomes (36). This is because each die has 6 possible outcomes (numbers 1 to 6), and when two dice are thrown simultaneously, the total number of outcomes is the product of the individual outcomes for each die.

Probability of getting a doublet = 6/36 = 1/6

A double of even numbers:
A double of even numbers is obtained when both dice show an even number. There are 3 even numbers on a die (2, 4, and 6), so the probability of getting an even number on one die is 3/6 = 1/2. Since the second die also needs to show an even number, the probability can be calculated by multiplying the probability of getting an even number on the first die with the probability of getting an even number on the second die.

Probability of getting a double of even numbers = (1/2)*(1/2) = 1/4

An even number on one die and a multiple of 3 on the other:
An even number can be obtained on one die in three ways (2, 4, and 6), and a multiple of 3 can be obtained on the other die in two ways (3 and 6). The probability of getting an even number on one die is 3/6 = 1/2, and the probability of getting a multiple of 3 on the other die is 2/6 = 1/3. The total probability can be calculated by multiplying these two probabilities.

Probability of getting an even number on one die and a multiple of 3 on the other = (1/2)*(1/3) = 1/6

Neither 9 nor 11 as the sum of the numbers on the faces:
The sum of the numbers on the faces of two dice can range from 2 to 12. To find the probability of neither 9 nor 11 as the sum, we need to subtract the probability of getting 9 or 11 from 1 (as all events together must cover the entire sample space).

The pairs of numbers that sum up to 9 are (3,6), (4,5), (5,4), and (6,3), which give us 4 favorable outcomes. The pairs of numbers that sum up to 11 are (5,6) and (6,5), which give us 2 favorable outcomes. Therefore, the probability of getting 9 or 11 is 4/36 + 2/36 = 6/36 = 1/6.

Probability of neither 9 nor 11 as the sum = 1 - 1/6 = 5/6

A sum less than 8:
To find the probability of getting a sum less than 8, we need to calculate the number of favorable outcomes (pairs of numbers that sum up to less than 8) and divide it by the total number of outcomes.

The pairs of numbers that sum up to less than 8 are (1,1), (1,2), (1