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Expected Value
            CSCE 115
 Revised Nov. 29, 2004, May 2, 2005
                                                  2

Probability
   Probability is determination of the chances
    of picking a particular sample from a
    known sample.

   Notation:  if A is some event, the
    probability of A is
                      P(A)
                                                   3

Probability
 Probability of  success (when the events are
  equally likely) is
        Number of successful outcomes
         Number of possible outcomes
 Example: If 1 student is picked at random from
  a class of 7 woman and 13 men, what is the
  probability that the student is a woman?
           P(woman) = 7/(13+7) = 7/20
                                                     4

Probability
   Non-example:      If you roll two dice and add
    the spots the possible outcomes are 2, 3, 4,
    5, 6, 7, 8, 9, 10, 11, 12. What is the
    probability of rolling a 2?
   Is the following correct?
    There is 1 success (getting a two) out of 11
    possibilities (2, 3, …, 12) so the probability
    is 1/11
   Why?
                                                  5

Independent events
   Two   events are independent if the way the
   first event happens does not affect the way
   the second event happens.
                                                       6

Example: Independent events
 Put 3 red balls and 2 green balls in a bag. Event
  1: select a ball at random from the bag and
  determine its color. Put the ball back.
  Event 2: Select a second ball at random.
  Events 1 and 2 are ____________
 Event 3: select a ball at random and set it aside.
  Event 4: select a second ball at random. Events
  3 and 4 are ____ _____________
                                                7

First fundamental rule:
   The probability that something does not
    happens is 1 - the probability it happens
              P(not A) = 1 - P(A)
   Example: The probability of picking a man
    from the class of 7 women and 13 men is
       1 - (7/20) = 20/20 - (7/20) = 13/20
                                                  8

Second fundamental rule
   Iftwo events are independent, the
    probability that both A and B happen is
           P(A and B) = P(A) * P(B)
   Example: We randomly select a ball from a
    bag with 3 red and 2 green balls. We put it
    back and draw again. The probability that
    both balls are red is
   P(red, red) = (3/5) * (3/5) = 9/25
                                                        9

Example: Role 2 dice
 Suppose   that we have a red die and a blue die. We
  roll and sum. What are the possible outcomes?
    RB RB RB RB                 RB RB
      1,1 1,2 1,3        1,4     1,5 1,6
      2,1 2,2 2,3        2,4     2,5 2,6
      3,1 3,2 3,3        3,4     3,5 3,6
      4,1 4,2 4.3        4,4     4,5 4,6
      5,1 5,2     5,3    5,4     5,5   5,6
      6,1 6,2     6,3    6,4     6,5   6,6
                                                  10

Example: Role 2 dice
 P(2) = 1/36               P(8) = 5/36
 P(3) = 2/36 = 1/18        P(9) = 4/36 = 1/9
 P(4) = 3/36 = 1/12        P(10) = 3/36 = 1/12
 P(5) = 4/36 = 1/9         P(11) = 2/36 = 1/18
 P(6) = 5/36               P(12) = 1/36
                P(7) = 6/36 = 1/6
                                                                 11

Probability of rolling 2 dice

                  0.2
  Probability.



                 0.15
                  0.1
                 0.05
                   0
                        2   3   4   5   6   7   8   9 10 11 12
                                Sum of two dice
                                              12

A children's game with spinner
The spinner is used to
                               3    3
determine how far you      7            5
move. What is the
probability of each       3             2
move?
P(2) = 1/10                8            6
P(3) = 4/10
                               3    6
P(5) = 1/10
                   How far, on the average,
P(6) = 2/10
P(7) = 1/10        do you expect to move
P(8) = 1/10        each time you spin?
                                                   13

A children's game with spinner
 Just averaging  the possible the possible outcomes
        (2 + 3 + 5 + 6 + 7 + 8) = 31 = 5.1667
                     6               6
  is _____ correct because the various values are
  not equally likely.
 A correct way is
  (2 + 3 + 3 + 3 + 3 + 5 + 6 + 6 + 7 + 8) = 46
                      10                     10
  = 4.6
                                                14

  A children's game with spinner
 (2 +3 + 3 + 3 + 3 + 5 + 6 + 6 + 7 + 8)
                   10
 = (2 +3*4 + 5 + 6*2 + 7 + 8)
                 10
= 2 * 1 + 3 * 4 + 5 * 1 + 6 * 2 + 7 * 1 + 8 * 1
        10      10       10       10     10    10
 = 2*P(2) + 3*P(3) + 5*P(5) + 6*P(6) + 7*P(7)
            + 8 * P(8)
                                                     15

Expected value
 Suppose that a certain experiment X could result
 in the values of {a, b, c, …, k} and the
 probabilities of these outcomes are P(a), P(b),
 P(c), …, P(k). The expected value is
 E(X) = a * P(a) + b * P(b) + c * P(c)
                                    + … + k * P(k)
                                            16

Example: Expected value
   Recall   the kid's spinner game
  P(2)  = 1/10         P(3) = 4/10
    P(5) = 1/10         P(6) = 2/10
    P(7) = 1/10         P(8)= 1/10
  E(spin) = 2 * .1 + 3 * .4 + 5 * .1
               + 6 * .2 + 7 * .1 + 8 * .1
            = 4.6
                                                     17

Example: Expected value
   Experiment:    You flip a coin and get 1 point
    of a head and 0 points for tail
   P(head) = .5,    P(tail) = .5
   E(flip) = 1 * .5 + 0 * .5 = .5
                                               18

Example: Expected value
   Experiment:   You roll one die and count
    dots.
   P*(1) = 1/6, P(2) = 1/6, … P(6) = 1/6
   E(roll) = 1*(1/6) + 2*(1/6) + 3*(1/6) +
    4*(1/6) + 5*(1/6) +6 *(1/6)
  =      1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
    = 21/6 = 3.5
                                                     19

Example: Expected value
  Experiment:  You roll two dice and count dots
  P(2) = 1/36      P(6) = 5/36       P(9) = 4/36
   P(3) = 2/36      P(7) = 6/36       P(10) = 3/36
   P(4) = 3/36      P(8) = 5/36       P(11) = 2/36
   P(5) = 4/36      P(12) = 1/36
  E(two dice) =
   2*(1/36) + 3*(2/36) + 4*(3/36) + 5*(4/36)
   6*(5/36) + 7*(6/36) + 8*(5/36) + 9*(4/36)
   10*(3/36) + 11*(2/36) + 12*(1/36)
  = 252/36 = 7
                                                     20

Example: Expected value
 John proposes    that the charity bazaar sell tickets
 for $2. The player rolls 2 dice. The play wins $12
 if the roll is a 2 or a 8. On the average, how much
 will the charity win each time a player rolls the
 dice?
                                                      21

Example: Charity bazaar
 Solution  1:
 Outcomes are P(2) = 1/36,      Prize(2) = $12
                   P(8) = 5/36, Prize(8) = $12
               P(other) = 30/36, Prize(other) = $0
  E(Prize) = $12*(1/36) + $12*(5/36) +
                    $0*(30/36) = $72/36 = $2
 Cost of ticket - Expected value of prize
  = $2 - $2 = 0
 The charity does not expect to win any money with
  this game.
                                                  22

Example: Charity bazaar
   Solution 2:
   Outcomes   are charity wins $2 or loses $10
   P($2) = 30/36 = 5/6
    P(-$10) = 1/36 + 5/36 = 6/36 = 1/6
   E(winnings) = $2 * (5/6) + (-$10) * (1/6)
    $10/6 + (-$10/6) = 0
                                                        23

Fair Game
   A fair  game is a game where the expected value
    of winning is 0
   Fair games are highly desirable when play with
    your friends
   Fair games are not desirable for organizations
    trying to earn money by offering games of chance.
   Casinos would go out of business if they had fair
    games.
                                                          24

Example: Charity bazaar
 John  proposes that the charity bazaar sell tickets for $2.
  The player rolls 2 dice. The player wins $12 if the roll is
  a 2 or a 11. They win nothing otherwise. On the
  average, how much will the charity win each time a player
  rolls the dice?
 P(-$10) = 1/36 + 2/36 = 3/36 = 1/12
  P($2) = 1- 1/12 = 11/12
 E(winnings) = (-$10) * (1/12) + ($2)*(11/12)
                = (-$10 + $22)/12 = $12/12 = $1
 The game is ___ fair. This is desirable for _____.
                                                                  25

Odds
   Unfortunately   popular slang uses “odds” in at
    least 3 different ways.
   It may indicate the payoff if you win a bet
   It may be a synonym for probability
     – This is used by the Washington State Lottery
     – It is used in many popular articles about odds
     may indicate the ratio of the probability of
   It
    winning to the probability of losing
     – This is the definition we will use
     – This the definition is the one normally sees in math and
       statistics books
                                                   26

Odds of Winning
   Suppose the  probability of winning is p and
    probability of losing is q = 1 - p. Then the
    odds of winning are p:q.
   We treat p:q as a fraction and normally
    multiply and divide both parts to clear of
    fractions and to remove common factors.
   The odds of losing are q:p.
                                               27

Example: Odds
 In therevised charity bazaar game, the
  probability of a player winning is 1/12.
 The probability of losing is ____
 The odds of winning are 1/12:11/12 or ____
 The odds of losing are ______
                                                28

Example: Odds
   You rolltwo dice and win if you roll a 9.
   The probability of winning is 4/36
   The probability of losing is ____
   The odds of winning are 4/36:32/36
     = 4:32 = ___ : ___
                                                29

Example: Odds
 The odds of winning a game are 5:31. What is the
  probability of winning and losing?
 Suppose that you played 5+31 = 36 times. You
  would expected to win 5 times and lose 31 times.
 The probability of winning is 5/36.
  The probability of losing is 31/36.
                                                       30




 Suppose   the odds of winning are given as a:b. We
  want to the probability of winning.
 Algebraically, suppose p:q = p:(1-p) = a:b
   Treat ":" like it was a "/“
 p/(1-p) = a/b
 bp = a(1-p)
 bp = a - ap
 ap + bp = a
 (a+b)p = a
 p = a/(a+b)
                                                      31

Example: Odds
   The odds of winning first prize in a raffle are
    1:1999. What is the probability of winning?
   Suppose that 1+1999 = 2000 tickets are sold. We
    would expect to win 1 time out of 2000
   The probability of winning is 1/2000.
   Using the formula: a = ___, b = _____
   The probability is p = a/(a + b)
                         = 1/(1 + 1999)
                         = _________
                                                                    32

 Example: Raffle
 The prize list for a raffle is
 Prize              Number       Value           Odds
  New car (Kia)           1      $10,000         1:9999
  TV set                 10         $300          1:999
  Meal for two           20          $50          1:499
 Determine:
   – Expected value of a ticket
   – Number of tickets sold
   – If all of tickets costing $2.50 each are sold and if there is an
     addition cost of $2000 for printing and advertising, write a
     budget for the sponsors
   – Solution: See
     http://www.cs.plu.edu/courses/csce115/spring05/download/Pro
     bability/Raffle.xls
33

				
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