Probability and Expected Value
6.2 and 6.3
Expressing Probability
• The probability of an event is always between
0 and 1, inclusive.
• A fair coin is tossed once, what is the
P(heads)?
• What does a probability of 0 mean?
• What does a probability of 1 mean?
Theoretical Method
for Equally Likely Outcomes
•A card is drawn from a standard shuffled deck.
What is the probability that the card is a king?
•What is the probability that the card is a heart?
•A couple is expecting their fourth child. The
first 3 children are girls. What is the probability
that the fourth child is a boy?
Example
• In a recent year, 389 of the 281, 421,906 people
in the United States were struck by lightning.
Estimate the probability that a randomly selected
person in the United States will be struck by
lightning this year.
• In a study of brand recognition, 831 consumers
knew of Campbell’s Soup and 18 did not.
Estimate the probability that a randomly selected
consumer will recognize Campbell’s Soup.
Complements
• The complement of an event consists of all
outcomes in which the event does not occur.
• What is the complement of the event “a rainy
day”?
• What is the complement of the event “at least
one girl” in a four child family.
• P(not A) = 1-P(A)
• If there is a 30% chance of rain today, what is
the probability of no rain?
• What is the probability of at least one girl in a
four child family?
Probability Distribution
• A probability distribution represents the
probabilities of all possible events.
1. List all possible outcomes.
2. Group outcomes into the events you are
concerned with.
3. Find the probability of each event.
4. List in a table. Be sure the sum of the
probabilities is 1.
Example
• Find a probability distribution for the number
of girls in a 3 child family?
The Law of Large Numbers
• Applies to independent events—repeated
trials to do not depend on earlier trials.
• If the process is repeated many times, the
proportion of trials in which event A occurs
will be close to the P(A). The larger the
number of trials, the closer the proportion
should be to P(A)
Experiment tossing a Coin
Toss a coin the indicated number of times and
record the proportion of heads.
# of tosses
Proportion of
heads
# of tosses
1
30
5
40
10
50
20
60
Proportion of
heads
What happens to the proportion of heads as the
number of tosses increases?
Expected Value
• The expected value of a variable is the
weighted average of all its possible events.
• Only find the expected value when there are a
large number of events.
Thought Question 1
Suppose that a sorority is selling raffle tickets to raise
money. The grand prize is a 32gb iPod Touch valued
at $280, and the members must sell all 1000 raffle
tickets that were printed. How much would you be
willing to pay for a single ticket? Explain your
answer.
How much should the
sorority charge if they want
people to buy the tickets,
but still make a profit?
IPod Touch Expected Value
• Suppose the sorority thinks they can sell all
1000 tickets for $1 each. Remember the value
of the IPod touch is $280. Find your expected
value if you choose to buy a ticket?
Deal or No Deal?
(1) You choose one of four sealed cases; one contains
$1,000, and the others are empty. If you open your
case, you have a 25% chance to win $1,000 and a
75% chance of getting nothing (winning $0).
(2) Or, you can sell your unopened case for $240,
giving you a 100% chance of winning $240.
(3) Find the expected value of each situation?
(4) Will you open or sell your case?
Life Insurance
The CNA Insurance Company charges a 21-year-old male a
premium of $250/year for a$100,000 life insurance policy. A
21-year-old male has a 0.9985 probability of living for a year
(based on data from the National Center for Health Statistics.
• From the perspective of a 21-year-old male (or his estate),
what are the values of the two different outcomes?
• What is the expected value for a 21-year-old male who buys
the insurance?
• What would be the cost of the insurance policy if the
company just breaks even (in the long run with many such
policies), instead of making a profit?
• Should a 21-year-old make buy a $100,000 life insurance
policy?
Gambler’s Fallacy
• The gambler’s fallacy is the mistaken belief
that a streak of bad luck makes a person “due”
for a streak of good luck.
The Law of Large Numbers
Gambling
The “house” in a gambling operation is not
gambling at all
– The games are defined so that the gambler
has a negative expected gain per play.
– Each play is independent of previous plays, so
the law of large numbers guarantees that the
average winnings of a large number of
customers will be close to the (negative)
expected value.
1
6
If You Can Play Tic-Tac-Toe,
You Can Win Lucky Dough!
• You can win cash with the Lottery’s newest Numbers game, Lucky
Dough. This new game, along with Club Keno, is sold only in Lottery
locations with social environments and a liquor-by-the-drink license. This
fun new game offers:
• A growing top prize that starts at $10,000 and grows until someone wins it!
• Average winning jackpots of $50,000 with the potential to grow to more
than $200,000.
• Seven different prize levels!
• A drawing every 5 minutes!
• Great chances to win at 1 in 6.01!
• A play action that is as easy as tic-tac-toe!
Use the information on the website to compute the expected value.
http://www.molottery.com/lucky_dough/lucky_dough.shtm
Probability Distribution for Lucky Dough
Winnings
$10,000
$200
$100
$50
$15
$5
$2
?
Probability
1/390625=0.00000256
1/24414=0.00004096
1/8138=0.0001229
1/2035=0.0004914
1/313=0.003195
1/59=0.01695
1/7=0.14286
?