7.4 Odds,
Conditional Probability and Expected Value
|
Odds in Favor: |
Odds Against |
|
Success: |
Failure: |
Odds of 12:1 means 12 chances will happen (success)
1 change that it won't (failure)
What odds in favor are equivalent to Probability of 12/13
I prefer odds notation using :
Odds
P(A):P(not A); or n(A): n(not
A)
**n(A) is notation
that means the number of elements in set A
On roll of pair of die, what are the odds for rolling a
7? Odds against rolling a 7?
Draw a card from a deck of cards, find odds for a club? Odds against a club?
If 3 out of 4 marriages last at least 20 years
What are odds
for marriage to last at least 20 years?
What are odds
against marriage lasting 20 years?
Family of 4 children, what are odds that all are same sex?
If odds against Sam winning the fight are 3 to 5, what is
the probability that he'll win?
Death Table - Gives odds against dying in the next year
|
Age |
Male |
Female |
|
15-24 |
576-1 |
1814-1 |
|
25-34 |
560-1 |
1489-1 |
What is the probability that a female from 25-34 will live
another year?
Johnny bats 7 times, hits 3 - Find probability of a hit,
odds of a hit, odds against a hit
If P(A)= 4/9, What are odds for event A? Against event A?
If odds in favor of event E are 2:3, what is probability
that E occurs?
If odds against are 1:15, what is probability that happens?
Conditional Probability
“An
additional condition is known”
Essentially, the “given event”
becomes the new, reduced sample space.
Example: In a family of three children, find the
probability that the family has exactly two boys, given the first child is a
boy.
S = {BBB, BBG, BGB, BGG, GBB, GBG,
GGB, GGG}
A = 1st child a
boy = {BBB, BBG, BGB, BGG} (new
sample space)
B = exactly two boys = { BBG, BGB,
GBB}
A
B = 1st child a boys AND exactly two boys = { BBG,
BGB} (new event)
Probability in a family of three
children there are exactly two boys, given first a boy
P(B|A)
=
If A and B are events in sample space
S and 
, then the conditional probability that event
B occurs given that event A has occurred is given by
Example: If you toss two fair dice and consider the
sum of the up faces.
a) What is the probability that the sum is 11?
b) What is the probability that the sum is an
11, given you know that the sum is greater than 10?
Example: If 
, find
a) P(B|A)
b) P(A|B)
Expected Value Book
reads well - formula page 451
Expected value is average of winnings over long run
Consider the spinner show with the payoff in each
section. Find the probability associated
with each range.
|
Outcome |
P(Outcome) |
Outcome *P(outcome) |
|
$1 |
|
|
|
$2 |
|
|
|
$3 |
|
|
|
$4 |
|
|
Find the expected value of the payoff for spinning the
spinner once?
What would be a fair price to pay to play the game?
Fair Game
A game is considered “fair” if and only if its expected value equals the
price per game. (Nobody wins, nobody
loses)
Example: An insurance company - insure
dorm room against theft
Value of
possessions $800
Probability
of $400 robbery is 1/100, Probability of $800 robbery is 1/400
How much should insurance company charge to cover costs and
make $25 profit/premium?
Example: A dice game
which has no charge to play. Roll a
single die, and the payoff is
$2 for rolling a 6
$1 for rolling a 5
$0 for rolling a 4
You pay $1 for rolling a 3
You pay $2 for rolling a 2
You pay $3 for rolling a 1
Find the expected value of the
game. Is it fair?
Example:
Ten thousand raffle tickets are sold at $2 each for a benefit. Prizes are awarded as follows: 2 prizes of $1,000; 4 prizes of $500, and 10
prizes of $100. What is the expected
value of the raffle if you purchase 1 ticket?
Consider the payoff – disregard $2
purchase cost
Consider the net result –including
the $2 purchase cost
Assignment: Homework Math XL for section 7.4 and
textbook pg 482 – 3, 6, 7, 11, 13-18, 21, 30
.