4 math 230 february 2013 midterm exam 1 question 3 20

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Math 230
February 2013
Midterm Exam 1
Question 3 (20 pts) . This question has two parts.
a) Suppose that 13 cards are dealt without replacement from a standard deck of 52 cards.
(i) What is the probability that exactly three of the cards will be spades?
(ii) What is the probability that six cards will be of one suit, and the remaining seven will
be of another suit?
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Math 230
February 2013
Midterm Exam 1
(iii) What is the probability that six cards will be of one suit, six cards will be of a 2nd suit
and the remaining one card will be from a 3nd suit? (Notice how this is di ff erent from
the previous question)
b) Now suppose that 4 cards are dealt with replacement from a standard deck of 52 cards. What
is the probability that there will be at least two jacks among the 4 cards?
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Math 230
February 2013
Midterm Exam 1
Question 4 (15 points) . Let X have mean a and Y have mean b . Let V ar ( X ) = σ
2
, E ( Y
2
) = s
2
.
a) For this part assume that X and Y are independent random variables. If Z =
1
3
X − Y , find
V ar ( Z ), and E XZ .
b) If we now assume that X and Y are no longer independent random variable and that c =
E [ XY ]. If in addition Y > 0 then define the function K by K ( m ) = E [ Y ( X − m )
2
].
(i) Find the value of m which minimizes K ( m ).
(ii) Why does your result give the expected answer if X and Y are now assumed to be
independent ?
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Math 230
February 2013
Midterm Exam 1