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2.2. A red die and a blue die are rolled and the number on each is recorded.

a) List the elements of the sample space.

b) List the elements in the following events:

• A: the sum of the two numbers is at least 9,

• B: the blue die has a larger value than the red die,

• C: the blue die is a 5.

c) List the elements of the following events: An B, BUC, AN (BUC).

2.8. Five cards are dealt from a standard 52-card deck. What is the probability

that the hand contains three of a kind?

(Three of a kind means that there are 3 cards of one denomination, 1 of another,

and 1 of a third. Example: 266, 27, 29, 70, 104. Note that by this definition a

full house is not three of a kind.)

=

2.18. Let Aį be any events.

a) Show that

P(A1 n A2 n A3) = P(AL NA2 n A3)

= P(A1).P(A2 | A1). P(A3 | A1 n A2).

b) Generalize this to give a formula for the intersection of any number of events.

[Hint: Use induction.]

2.21. A survey was given to a class of students. One of the questions had two

versions – a) and b) below – which were assigned to the students randomly:

Social science researchers have conducted extensive empirical studies and

concluded that the expression

a) “absence makes the heart grow fonder”

b) “out of sight out of mind”

is generally true. Do you find this result surprising or not surprising?

Here are the results.

А AC

absence sight

6

5

44 36

surprised (S)

not surprised (SC)

Is “being surprised” independent of the form of the question asked?

2.21. Use the definition of independent events to show that

a) if A and B are independent, then A and B are also independent;

b) if A and B are independent, then AC and B° are also independent.

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