Section 7.3 Contd

1
Section 7.3 (Contd)

• Familiarizing ourselves with probabilities

2
Goals for the day

• Finish computing the probability of rolling a
  certain sum of two dice.
• Computing Expected Values
• Independence vs. dependence
• Additional probability problems
• Considering all possibilities
• Determining the right probability to compute

3
Finishing up the dice rolls

• d4 d10 d6 d8

4
Expected Value

• Expected value is the outcome, on average, you
  expect to occur.
• If the possible outcomes are X(1), X(2), X(n)
  each with probability P(1), P(2), P(n)
  respectively, then the expected value is
• E X(1)P(1) X(n)P(n)

5
Determining Expected Value

• As you compute the expected value, consider the
  following
• Explain the definition of expected value.
• If the values X(i) are payouts or losses of a
  game, explain how the expected value can be used
  to determine if the game is fair or not.

6
Independence

• Compute the expected values of rolling a d4, d6,
  d8, and d10.
• Do these numbers relate to the expected values
  you previously computed? If so, how?

7
What is dependence?

• Two events are dependent if the probability of
  one is influenced by the other.

8
An example of dependence

• A bag has 20 bills in it, one is a 50 bill, and
  the rest are 1.
• What is the probability of drawing the 50 bill
  after two draws.
• What is the expected value of drawing two bills
  from the bag?
• What is the expected value of the second draw?

9
Expected Values in Games

• The state expects to sell 250,000 1 tickets of a
  new lottery where one winner is selected at
  random to draw the blue ball from a bag with one
  blue and four red balls
• You win 1,000,000 if you pull the blue ball
  first
• You win 100,000 if you pull the blue ball second
  
• You win 10,000 if you pull the blue ball third
• You win 1,000 if you pull the blue ball fourth
• You win 100 if you pull the blue ball fifth
• Can the state expect to make a profit on this
  game?

10
Exploring the Lottery

• If this is not finished by class, it will be
  homework due next Thursday, in addition read
  section 7.4
• http//www.arizonalottery.com/Powerball/howtoplay.
  asp

11
An example

• Compute the expected value of buying one ticket
  of PowerballTM at 1.
• How much should a ticket cost in order for it to
  be a fair game?