STAT 110 - Section 5
Lecture 23
Professor Hao Wang
University of South Carolina
Spring 2012
Example on tree diagram
There is an 80% chance you will make the first red
light driving in to work.
If you make the first, there is a 90% chance you will
also make the second.
If you miss the first, there is a 90% chance you will
also miss the second.
What is the chance you get stopped by both lights?
A) 0.08 = 8%
D) 0.18 = 18%
B) 0.09 = 9%
E) 0.72 = 72%
C) 0.16 = 16%
Example on sampling distribution
A simple random sample of 1000 college students is
taken and asked if they agree that “The dining choices
on my campus could be better.
If the true percent for the whole population is 60%, the
percent in the sample who will say “Yes” should be close
to a normal distribution with mean 0.6 and standard
deviation of about 0.015.
What’s the probability that 0.63 or more in the sample will say
“Yes”?
A 2.5%
B 5%
C 10%
D 60%
E 1.5%
Chapter 20: Expected Value
How much is a
Pick 4 lottery
ticket worth?
In a Pick 4 game, you make a Quick-Pick $1
Straight-Box wager.
The computer chooses a four digit number, and you
have two ways to win.
1. You win $3,100 if you exactly match the
winning number.
2. You win $600 if your number has the same
four digits as the winning number, in any
order.
The probability model according to their web-site is:
outcome
$0
probability 0.9996
$600
0.0003
$3,100
0.0001
expected value – found by multiplying each
outcome by its probability and
then summing over all possible
outcomes
possible outcomes: a1, a2, …, ak
probabilities: p1, p2, …, pk
expected value = a1 p1 + a2 p2 + … + ak pk
outcome
$0
probability 0.9996
$600
0.0003
$0 + $0.18 +
$3,100
0.0001
$0.31 = $0.49
•
Suppose a sorority is selling 1000 raffle tickets for
$1.00 each. One ticket will be drawn at random
and the winner receives $200.00. There are two
possible outcomes here:
1.
Win $200.00 (so net gain is $199 with probability
1/1000)
2.
Do not win (so net “gain” is -$1 with probability
999/1000)
So how would we find the average net gain for $1
ticket?
• An insurance company sells a particular
policy that costs the insured $1,000 to
purchase. The insurance company knows
that 1/100 of these policy holders will file a
claim worth $20,000, 1/200 will file a $50,000
claim, and 1/500 will file a claim of $100,000.
Find the insurance company’s expected net
gain for one such policy.
You play a game with two possible outcomes.
Outcome A has probability 0.4 and outcome B has
probability 0.6. When B occurs you win $2.00;
otherwise, you lose $1.00. What is your expected
value for this game?
A. $2.00
B. $0.80
C. $0.20
D. -$0.80
• Suppose we have the following distribution of
outcomes for a “loaded” die.
• Outcome 1
2
3 4 5
6
• Probability 0.5 0.1 0.1 0.1 0.1 0.1
• Find the expected number you’ll get when
rolling this die.
A.1 B. 1.5 C. 2 D. 2.5 E. 3
Expected value and house edge
Roulette. Say we bet $10 on Black.
There are, 0 to 36, so 37 possible outcomes. The
possible outcomes are anyone of 18 possible
Blacks, or anyone of 18 possible Reds, Or Green
(don't forget that with roulette the zero is Green).
1.
2.
Doubling money if black (so net gain is $10 )
Losing money otherwise (so net “gain” is -$10)
What’s the chance of doubling money ?
A 1/37 B 36/37 C 19/37 D 10/37 E 18/37
Roulette. Say we bet $10 on Black.
There are, 0 to 36, so 37 possible outcomes. The
possible outcomes are anyone of 18 possible
Blacks, or anyone of 18 possible Reds, Or Green
(don't forget that with roulette the zero is Green).
1.
2.
Doubling money if black (so net gain is $10 )
Losing money otherwise (so net “gain” is -$10)
What’s expected (average) net gain for a $10 bet ?
A $-10/37 B $10/37 C $1/37 D $-1/37 E $18/37
Roulette. Say we bet $10 on Street.
There are, 0 to 36, so
37 possible outcomes.
1.
2.
Win $120 if any three numbers horizontal (so net gain is
$110 )
Lose money otherwise (so net “gain” is -$10)
What’s the chance of winning ?
A 1/37 B 33/37 C 19/37 D 3/37 E 12/37
Roulette. Say we bet $10 on Street.
There are, 0 to 36, so
37 possible outcomes.
1.
2.
Win $120 if any three numbers horizontal (so net gain is
$110 )
Lose money otherwise (so net “gain” is -$10)
What’s expected (average) net gain for a $10 bet ?
A $-10/37 B $10/37 C $1/37 D $-1/37 E $18/37
The Law of Large Numbers
law of large numbers  If a random phenomenon
with numerical outcomes is repeated many times
independently, the mean of the actually observed
outcomes approaches the expected value.
• In many independent repetitions,
- the proportion of each possible outcome will
be close to its probability.
- the average outcome obtained will be close
to the expected value.
The Law of Large Numbers
Application
• The Law of Large Numbers explains why
gambling is a business for a casino – they use
the fact that the average winnings of a large
number of customers will be quite close to its
expected value and that is how they make
money
• Other areas: Insurance, lottery, etc