Transcript probability

Sullivan Algebra and
Trigonometry: Section 14.3
Objectives of this Section
• Construct Probability Models
• Compute Probabilities of Equally Likely
Outcomes
• Utilize the Addition Rule to Find Probabilities
• Compute Probabilities Using Permutations and
Combinations
An event is an outcome from an
experiment.
The probability of an event is a measure
of the likelihood of its occurrence.
A probability model lists the
different outcomes from an
experiment and their corresponding
probabilities.
To construct probability models, we need
to know the sample space of the
experiment. This is the set S that lists all
the possible outcomes of the experiment.
Determine the sample space resulting from
the experiment of rolling a die.
S = {1, 2, 3, 4, 5, 6}
The probability of each outcome in the sample
space S = {e1, e2, …, en} has two properties:
1. 0  P(ei )  1 for all events ei
The probability assigned to each outcome
is non-negative and at most 1.
n
2.  P(ei )  P(e1 )  P(e2 )   P(en )
i 1
=1
The sum of all probabilities equals 1.
Probability for Equally Likely Outcomes
If an experiment has n equally likely
outcomes, and if the number of ways an
event E can occur is m, then the
probability of E is
Number of ways that E can occur m
P( E ) 

Total number of possible outcomes n
A classroom contains 20 students: 7 Freshman,
5 Sophomores, 6 Juniors, and 2 Seniors. A
student is selected at random. Construct a
probability model for this experiment.
7
P( F ) 
20
5
P( Soph) 
20
6
P ( Jr ) 
20
2
P ( Sr ) 
20
Theorem Additive Rule
For any two events E and F
P( E  F )  P( E )  P( F )  P( E  F )
P( E  F )  P( E )  P( F )
if E and F are mutually exclusive.
What is the probability of selecting an
Ace or King from a standard deck of
cards?
4 1
4 1
P(Ace) = 
P( King) = 
52 13
52 13
P(Ace or King)
= P(Ace) + P(King) - P(Ace and King)
2
1 1
  0 
13
13 13
Probabilities of Complementary Events
If E represents any event and E represents
the complement of E, then

P E  1  P( E )
Suppose the probability that a hurricane hits a county
in a given year is 0.02. Find the probability that a
hurricane doesn’t hit the county.
Since there are only two possible events in
the sample space, hurricane or no hurricane,
these events are complementary.
Prob(No H) = 1 - Prob(H) = 1 - 0.02 = 0.98
Suppose you managed a little league team.
You have 8 pitchers, 10 fielders, and 5 other
players on the bench. If you choose three
players at random, what is the probability that
they are all pitchers?
Prob(3 Pitchers) = # of ways to choose 3 pitchers
# of ways to choose 3 players
8! 8  7  6  5!

 8  7  56
8 C3 
3!5!
3  2  5!
23!
23  22  21  20!

 23 11  7  1771
23 C 3 
3!20!
3  2  20!
Prob(3 Pitchers) = # of ways to choose 3 pitchers
# of ways to choose 3 players
56

 0.0316
1771
or
3.16%