Probability
1.
You have a fair 4-sided dice. One side of the dice has the numb
er 1, another side has the number 2, another side has
the number 3, and another side has the number 4. You then con
sider the following random experiment: “Toss the 4-
sided dice twice, and record the number you obtained from each tos
s.” You then consider the following random
variable
X
: “The number obtained in the first toss plus the number o
btained in the second toss.”
(a)
Provide the sample space of the random experiment.
(b)
List all the possible values the random variable may take.
(c)
Sketch (by hand) a histogram of the distribution function
of the random variable.
(d)
Calculate the expected value of the random variable.
(e)
Calculate the variance and the standard deviation of the random var
iable.
(f)
Calculate P(
X >
4).
(g)
What is the 75
th
percentile of
X
?
(h)
Suppose that you conduct the experiment twice. What is the pro
bability that you obtain
X
= 3 in the first
experiment and then, conditional on that, you obtain
X
= 7 in the second experiment?
2.
Consider a lottery where you choose three integer numbers between
1 and 20. A number can only be used once
within a ticket. And, the order of the sequence of numbers
matters for the uniqueness of the ticket. For example, a
ticket comprised of the sequence 19, 7, 12 is different than th
e ticket 7, 12, 19. Suppose that 30 people play the
lottery.
(i)
What is the probability that there will be at least 3 winners
of the lottery?
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