Lecture 1: Solutions to the SelfTest .

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Lecture 1 Solutions to the Self-Test .
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1. For what number of students the probability of
at least 2 having the same BD is 0.7?
Solution. Lets consider first a complimentary
event A There arent any matching BDs in the
class of n students.
In what follows, I use the Mathematica
notations, so that you could learn some of M. in
parallel to solving the problem. Probn_ 1-
365!/((365-n)!365n) ( Here we build the
function Probn defining P(A) ) It may look
strange to you, until you notice that for any
number s gtk, the following is true s(s-1) (s-2)
(s-k1) s!/(s-k)!. You can also try the
smaller values of n (n2,3,4) to understand it
better.
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Mathematica woks very efficiently with the
tables. Let us create a table of probabilities
for the number of students, n, varying from 1 to
a maximum chosen value nmax. TabProbnmax_
Tablen,Probn,n,1,nmax We can also plot
this table ProbPlotnmax_ListPlotTabProbnmax
, PlotJoined-gtTrue, Frame-gtTrue,
GridLines-gtAutomatic Let us choose, for
example, nmax 50, and evaluate the function
ProbPlot50. If did everything correctly, you
will get a plot
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ProbPlot50
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2. A card is drawn at random from an ordinary
deck. Describe the sample space if   a. suit is
not considered   Answer Sample Space
consists of 13 cards, from the 2 through the
Ace.   b. if suit is considered Answer Sample
Space contains all 52 cards. 3. Referring to the
previous problem, let A"a king is drawn" and
B"a club is drawn". Describe the events a)AUB,
(b) AB, (c) AUBc, (d) Ac U Bc, (e)
(AB)U(ABc) Hint According to the definition of
a union, a point a belongs to the union of two
events C and D (CUD) if it belongs to at least
one (may be both) of the events . We can also
say that a does not belong to the union CUD if
it belongs to none of those events.
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• (Here I repeat the statement)
• Referring to the previous problem, let A"a king
  is drawn" and
• B"a club is drawn". Describe the events
• a) AUB, (b) AB, (c) AUBc, (d) Ac U Bc, (e)
  (AB)U(ABc)
• Answers
• AUB A king, or a club, or a king of clubs (16
  outcomes)Let us consider, for example,5H. Does
  it belong to this set? The answer is no,
  because 5H is neither a king nor a Club.
• The king of clubs (1) (a King and a Club at the
  same time)
• A king or not a club (40 outcomes 39 (3 suits)
  1 (King of Clubs) ) Again, consider 5H. Does
  it belong to this set? The answer is yes,
  because 5H is not a club. How about KC? It also
  belongs to this set, because (though it belongs
  to the Clubs) its also a King. Therefore, our
  set includes three suits that are not clubs(39
  total), and KC in addition. The total number of
  cards in this set is 40.
• Not a king or not a club All cards excluding
  the king of clubs (51)Try to prove that any card
  excluding KC satisfies at least one of these
  conditions.
• A a king is drawn.
• Notice that AB is exactly the King of Clubs,
  while ABc includes all other Kings.

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• Problem 4 Describe in words the events
  specified by the following subsetsof W
  HHH,HHT,HTH,HTT,THH,THT,TTH,TTT.
• AHHH,HHT,HTH,HTTthe first coin shows
  HeadsBHHH,TTT all outcomes the
  same.BHHT,HTH,THH two outcomes are
  HeadsBHHT,HTH,HTT,THH,THT,TTH,TTT at least
  one tail What are the probabilities of these
  events?
• P(A)1/2, P(B) 1/4, e.t.c.
• Problem 5. A die is loaded in such a way that the
  probability of each face turning up is
  proportional to the number of dots on the face.
  What is the probability of getting an even number
  in one throw?Let us designate p the probability
  of one dot p(1) p. Then, p(n) n p
  (probability of n dots is proportional to the
  number of dots) . We can find p from the
  condition that the total probability is 1
• (123456)p 21 p p1/21 P(even) p(2)
  p(4) p(6) 12/21.

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Problem 6 Let A and B be events such that
P(AB)1/4, P(Ac)1/3, and P(B) ½. What is
P(AUB)? P(AUB) P(A) P(B) P(AB)
1/22/3-1/411/12. Problem 7 A student must
chose one of the subjects, art, geology, or
psychology, as an elective. She is equally likely
to choose art or psychology and twice as likely
to choose geology. What are the respective
probabilities that she chooses art, geology, and
psychology? p(A)p(P) p, p(G) 2 p. 4 p1, p
¼. The answer ¼, ½, ¼.