# Question: consider a fair five sided dice a roll x from...

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Consider a fair. five sided dice; a roll X from a such a dice follows the probability distribution:

P(X = x), x belongs to {1,2,3,4,5}

Using this and properties of random variables, please answer the following questions.

1. Calculate the expected value E[X] of a roll of this dice. (1 Mark)

2. Imagine we roll a second five sided dice; call this roll Y.

(a) What is the expected value of X + Y as well as its variance (i.e., the sum of two five-sided dice rolls)? (2 Marks)

(b) What is the probability that X + Y > 8 ? (2 Marks)

3. If we roll one of our five sided dice three times and recorded the results, what is the probability that all three rolls will be greater than 3? (3 Marks)

4. If we roll one of our five sided dice three times and we are told that the first roll was an odd number. We now select (randomly) one of the three rolls, what is the probability that it is an odd number? (3 Marks)

5. If we roll one of our five sided dice five times. Find the probability that exactly three dice show the same number, (i.e., three of a kind), and the remaining two dice show the same number different from the previous number, (i.e., a pair). (2 Marks)

6. If we roll one of our five sided dice five times. Find the probability that three or more dice show the same number. (2 Marks)