Math 3350

1
Math 3350

• Dr. Sinn, Fall 2005
• September 8
• Chapter 3 Expectation

2
Example Overbooked Flight

• Airlines routinely overbook their flights.
  Suppose that, for a 50 seat flight, the airline
  typically books 55 passengers. The table below
  shows the probability that various numbers of
  passengers show up.

3
Probability Function Vocabulary

• Probability Distribution Function (pmf)
• Discrete rv
• Continuous rv
• Cumulative Distribution Functions (cdf)
• Probability Calculations
• P(Overbooked Flight) ??
• P(All who show up get seats) ??
• P(1st Standby Passenger gets seat) ??
• What is expected number of passengers per flight?

4
Roulette Probability Functions

• Pick Color / Parity Pays 2 1
• Pick Single Number Pays 36 1
• Pick Pair of Numbers Pays 18 1
• Pick Quartet of Numbers Pays 9 1

5
Research Question

• The Perfect Roulette Strategy
• Initial Bet 1
• Always pick red
• If red, pocket winnings, start over.
• If black, double last bet.
• Every time red appears, we win 1. Guaranteed.
• Why doesnt Vegas go broke?

6
Question for Next Time

• For the airline problem
• Suppose that each ticketed passenger represents a
  50 profit for the airline.
• Suppose each overbooked passenger represents a
  250 loss to the airline (in meals, hotels, extra
  tickets, etc).
• What is the best number of tickets to sell for
  each flight?

7
Charts Graphs

• Probability Distributions
• Probability Tables
• Line graphs
• Histograms
• Cumulative Distribution Histograms
• Probability Tables
• Line graphs
• Histograms

8
Question for Poker Journal

• Black Jack Dealer Rules
• Always hit a count of 16 or less
• Always stand on 17 or more (even soft 17s)
• What percentage of hands will the dealer go bust?
• Assume draws with replacement. Why?

9
Expectation

• Consider, on average, what we expect
• when we roll fair, 6-sided dice
• when we spin a roulette wheel
• when we flip a coin
• How can we mathematically express our
  expectations of probability simulations?
• Ideas?
• How is expectation different for continuous
  random variables compared to discrete?

10
Real-World Applications?

• Spin a 4-wedge spinner. The table on the next
  slide shows how much you win on each spin. Would
  you pay 2 to play?

11
Spinner 3 Game

• Table of Winnings
• Pay 2 to play.
• Spin Even No Pay-Off
• Spin 1 1 Pay-Off
• Spin 3 6 Pay-Off
• Should you play?
• Explain why or why not mathematically.

12
Expected Value

• Expected Value in Probability means the average
  outcome.
• Expected Value Mean of Probability Table
• E(X) µX
• If the mean is the expected value, what is the
  standard deviation?

13
Dice Example

• You design the following game of chance for your
  7 year old brother.
• Pay 1
• Roll a typical (fair) 6-sided die
• Should he pay 1 each roll to play given the
  following pay-offs?
• Roll Odd No Pay-Off
• Roll 2 or 4 1 Pay-Off
• Roll 6 5 Pay-Off

14
Dice Chart

• Table of Winnings Pay 1
• Roll Odd Pay Nothing (- 1)
• Roll 2 or 4 Pay 1 ( 0)
• Roll 6 Pay 5 (4)
• What is the Expected Value?
• Should he pay to play?

15
Morbid Life Insurance Example

• Suppose a life insurance company sells a 1-year,
  500,000 term life policy to a 50-year-old male
  for 2,900. Based on actuary tables, the
  probability he lives through the 1-year term is
  0.99433.
• Compute the expected value of this policy
• Interpret this value in context of the insurance
  companys bottom line