CHAPTER 14 WORKSHEET
We
are rolling two foursided dice having the
numbers 1, 2, 3, and 4
on their faces.
Outcomes in the sample
space are pairs such as
(1,3) and (4,4)
A)
How many elements are in
the sample space?
B) What is the probability that
the total showing is even?
C) What is the probability that
the total showing is greater
than six?
SOLUTIONS
A)
16
B) .5
C) 3/16
An
experimenter testing for
extrasensory perception has five
cards with pictures of a (s)tar, a
(c)ircle, (w)iggly lines, a (d)ollar
sign, and a (h)eart. She selects
two cards without replacement.
Outcomes in the sample space
are represented by pairs such as
(s,d) and (h,c).
A)
How many elements are in
this sample space?
B) What is the probability that a
star appears on one of the cards?
C) What is the probability that a
heart does not appear?
SOLUTIONS
A)
20
B) 2/5
C) 3/5
For
a)
the next problems;
Find the probability of the given
event.
b) Find the odds against the given
event.
FORMULA
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑
PROBABILITY FORMULA FOR COMPUTING
ODDS
If
E’ is the complement of the
event E, then the odds against E
are
P( E ' )
P( E )
QUESTIONS
A
total of three
shows when we roll
two fair dice.
SOLUTIONS

2
1
a) P ( E ) 

36 18

b) First find P(E’)
1
17
P( E ' )  1 

18 18
17
Then find
P( E ' ) 18 17


1
P( E )
1
18
We
draw a face card
when we select 1 card
randomly from a
standard 52-card
deck.
2)
a) P ( E )  12  3
52 13
b) 10 to 3
3
10
1
P( E ' )
10
13
13



3
3
P( E )
3
13
13
ASSUME THAT WE ARE DRAWING A 5-CARD
HAND FROM A STANDARD 52-CARD DECK.
What
is the probability that all
cards are face cards?
We
have to remember the counting
technique C(52,5) ways to select a
5-card hand from a 52-card deck.
COMBINATION
Def.
we choose r objects from a set
of n objects, we say that we are
forming a combination of n objects
taken r at a time.
 If
 Notation
C(n,r) = P(n,r) / r!
= n! / [r!(n-r)!]
C (12,5)
792

 0.00030473
C (52,5) 2,598,960
What
is the
probability that all
cards are red?
0.025
In
a given year, 2,048,861
males and 1,951,379 females
were born in the United
States. If a child is selected
randomly from this group,
what is the probability that it
is a female.
SOLUTION
Do
you remember how to solve
this problem?
Females
Females  Males
0.04878
You
are playing a game in
which a single die is rolled.
Calculate the expected value
for each game. Is the game
fair? See next slide for
question.
If
an odd number shows up, you
win the number of dollars
showing on the die. If an even
number comes up, you lose the
number of dollars showing on
the die.
1
P1  , V1  1,
6
1
P2  , V2  2,
6
1
P3  , V3  3,
6
1
P4  , V4  4,
6
1
P5  , V5  5,
6
1
P6  , V6  6
6
1  1
 1 
1

 1    2     3   ...    6 
6  6
 6 
6

1 2 3 4 5 6 9 12
1
       

6 6 6 6 6 6 6 6
2

The game is not fair.
You
are playing a game in which
a single die is rolled. If a four or
five comes up, you win $2;
otherwise, you lose $1.
0,
the game is fair.
For
the following problem, first
calculate the expected value of
the lottery. Determine whether
the lottery is a fair game. If the
game is not fair, determine a
price for playing the game that
would make it fair.
Five
hundred chances are
sold at $5 apiece for a raffle.
There is a grand prize of
$500, two second prizes of
$250, and five third prize of
$100.
1
P1 
, V1  495
500
2
P2 
, V2  245
500
5
P3 
, V3  95
500
492
P4 
, V4  5
500
NOW CALCULATE THE EXPECTED VALUE.
.99  .98  .95  4.92  2
$3
to make the game fair.