... until you have a girl. Or, stop after 2 ... Pr 1 girl

1
Multiplication Rules in ProbabilityExpected
Value Examples
2
Swedish Parking Example

• Two hour parking - no meters
• Police note clock position of air valve stems
  of two tires.
• Car is ticketed in two hours if valves still in
  same position

3
Ticketed driver claims

• Moved car and parked again in the same place.
• Valves end up in same positions
• Whats the probability of this?

4
Multiplication Rule

• Multiply probabilities to find the chance that
  several events occur in sequence (or at the same
  time)

5
The defendants calculation

• Assumed 12 equally likely clock positions for two
  independent tires
• (1/12) x (1/12) 1/144
• Judge let him off

6
A Family Planning Strategy

• Have children until you have a girl
• Or, stop after 2 boys
• Will this change the male-female mix in the
  population?

7
Probabilities for each outcome

• Girl first
• 0.5
• Boy, Girl
• 0.5 x 0.5 o.25
• Boy, Boy
• 0.5 x 0.5 0.25

8
Probability distribution number of boys

• Whats the average number of boys per family?

9
Expected value The long run mean

• Theoretical mean for a large sample or many
  repeats of a process
• Mean number of boys for a large population of
  families
• Calculations
• Make chart of values and probabilities
• For each value, calculate value x probability
• Total these to get expected value

10
Number of boys per family

• Expected value
• 0 0.25 0.50 0.75
• mean 0.75 boys per family for a large
  population of families

11
Distribution of girls per family

• Possible number of girls are 0 and 1
• Pr (0 girls) Pr (2 boys) 0.25
• Pr (1 girl) 1 - 0.75 0.25

12
Girls per family

• Expected Value Sum of (value x prob)
• 0 x 0.25 1 x 0.75 0.75
• Mean over many families is 0.75 per family

13
Average Family

• 0.75 boys
• 0.75 girls
• Sex mix of boys-girls stays equal

14
The Basic Principle

• Each independent birth has equal probability for
  boy and girl
• This basic probability dictates the overall mix

15
A roulette wheel

• 38 numbers
• 18 are red, 18 are black, two are green

16
Bet 1 on a number

• If right, you gain 35
• If wrong, you lose 1
• Is this a fair game?

17
Expected value of the bet

• Win 35 with probability 1/38
• Lose 1 with probability 37/38
• Expected value
• (35)(1/38) (-1)(37/38) -2/38 -0.05
• Not fair
• Over many bets, youll lose 5 cents per dollar bet

18
Fair Game Definition

• Expected Value 0
• In the long run, neither side gains
• Roulette be on number not fair since expected
  value not 0

19
Bet 1 on Black Number

• If right, win 1
• If wrong, lose 1
• Whats the expected value? Is this fair?

20
Expected value of bet on black

• Values and Probabilities
• Outcome 1, prob. 18/38
• Outcome -1, prob. 20/38
• Expected Value
• (1) (18/38) (-1)(20/38) -2/38 -0.05
• Not fair since Expected Value not 0