M362K (rusin) Spr2015 ---- HW 2

Please turn in on Thursday Feb 5.

Note: when typing math in an email like this, we use notations "a_n" to
indicate subscripts ("a sub n") and "a^n" to indicate exponents ("a to the n").


1. An experiment consists of rolling a die repeatedly until a "6" appears,
at which point the experiment is over.
(a) How would you describe the sample space?
(b) Let E_n denote the event that  n  rolls are necessary. Which elements
in your the sample space are contained in E_n?
(c) How would you describe the complement of this event? :
     E1 union E2 union E3 union ... union E99  

2. Suppose that  A  and  B  are mutually exclusive events for which
P(A) = 0.3 and P(B) = 0.5 . What is the probability that
(a) either A or B occurs?
(b) A occurs but B does not?
(c) both A and B occur?

3. A "poker hand" is any collection of 5 cards taken from a single, standard
52-card deck.
(a) How many poker hands are there?
(b) If the dealer gives you 5 cards at random from a well-shuffled deck,
what is the probability you will be given "four of a kind"? (That means
your hand will contain all four cards that bear a certain number, plus
one other card. For example this hand is a "four of a kind":
  the 4 of spades, the 4 of hearts, the 7 of hearts, 4 of diamonds, 4 of clubs )

4. Two fair dice are rolled, one at a time. What is the probability that
the second die will show a number higher than showed on the first die?

5. A forest contains 20 elk, of which 5 are captured, tagged, and released
(all at once). A week later, 4 of the 20 elk are captured. What is the
probability that 2 of these 4 have been tagged? Be sure to state any
assumptions you are making in your analysis.

6. An urn contains 12 (identical) red balls, 16 blue ones, and 18 green ones. 
Seven balls are drawn out of the urn at random. Find the probability that
(a) 3 red, 2 blue, and 2 green balls are withdrawn.
(b) All the withdrawn balls have the same color.

7. Suppose  n  balls are randomly distributed into  K  compartments. (Each
compartment can have any number of balls in it, including  0  and  n .)
Find the probability that the first compartment will have exactly  m  balls
in it. You may assume that all  K^n  arrangements are equally likely.

8. Show that if  E  and  F  are any events in any experiment, the
probability that  E  and  F  both occur is at least
  P(E) + P(F) - 1 
(This is called "Bonferroni's Inequality".)