Combinatorics and Probability
Section 13-3
Probability and Odds
HW# 49
•Section 13-3
•Pp. 856-857
•#13-29 odds, 37,41,42
Probability
• Then chance of something happening is called
probability.
• The set of all outcomes of an event is called a
sample space.
• A desired outcome is called a success
• Any other outcome is called a failure.
• The probability of an event is the ratio of the
number of ways an event can happen to the total
number of outcomes in the sample space.
Probability of Success and of Failure
• If an event can succeed in s ways and fail in f
ways, then the probability of success P(s) and
the probability of failure P(f) are as follows:
• P(s) = s
s+f
P(f) = f
s+f
Rem em ber … .
• An event that cannot fail (is certain) has a
probability of 1.
• An event that is impossible has a probability
of 0.
• Everything else falls between 0 and 1
0%
25%
50%
75%
100%
• 0
.25
.5
.75
1
• impossible unlikely
probable
likely
certain
Examples
• Before 2oo1, the United States had 42 presidents including
William Jefferson Clinton. Only two of them (George
Washington and John Adams) belonged to the Federalist
Party. If you were to select a president at random for a history
report, what is the probability that your selection would be a
federalist? P(federalist) = 2/42 = 1/21 or 4.7% Very
unlikely.
• A sock drawer contains single, unmatched socks; 19 are
white, 8 are black and 5 are blue. What is the probability that
a sock selected at random will be black? P(black) = 8/32 = ¼
or 25% probability.
• What is the probability that a sock selected at random will
NOT be white? P(NOT white) = 13/32 = 40.6%
Example
• A box of 60 baseball cards contains 8 cards
with print errors on them. If 5 cards are
selected at random, what is the probability
that all 5 have print errors?
• There are C(8,5)= 56 ways of selecting 5
defective cards. And there are
C(60,5)=5461512 ways of selecting 5 cards
• P(5 defective cards) = 56/5461512 = 1/97527
Complements
• P(s) and the P(f) are called complements
because their sum is 1.
• P(s) + P(f) = 1
Odds
• There is a difference between probability and
odds. Probability represents the chance of one
outcome. Odds represents a comparison of the
possible outcomes.
• The odds of the successful outcome of an event is
the ration of the probability of its success to the
probability of its failure.
• Odds= P(s)
•
P(f)
Example
• A gam e contains 24 cards that say “M ove ahead
one space” and 10 cards that say “M ove back one
space”.A card m ay be chosen random ly from
anywhere in the deck of 34 cards.
• W hat is the probability ofdraw ing a “M ove ahead”
card? P (ahead) = 24/34 = 12/17
• What are the odds that a player will move his or
her token ahead one space on any given turn?
We know that P(s) = 12/17 so the probability of
failure = 5/17 Thus the odds 12/5 12 to 5 in favor
of moving ahead.
Another example!
• Four male and 2 female students have been selected as equal
qualifiers for two college scholarships. If the awarded recipients are
to be chosen at random, what are the odds that one will be male
and one will be female?
• C(4,1) = 4 (number of groups of 1 male)
• C(2,1) = 2 (number of groups of 1 female)
• Using the basic counting principle we can find the number of
groups of 1 male and 1 female
C(4,1) x C(2,1) = 8 possible groups. Now the total number of groups
of 2 recipients out of 6 who qualify is C(6,2) = 15. So the number of
groups that do not have 1 male and one female is 15 – 8 = 7
So P(s) = 8
P(f) = 7
Odds = 8/7
15
15