Announcements
Finite Probability
Monday, October 3rd
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MyMathLab 4 is due Wednesday Oct 5
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Problem Set 4 is due Friday Oct 7
Today: Sec. 6.2: Assignment of Probabilities
Create a probability distribution for an experiment
Calculate the probability of an event using set operations
Next Class: Sec. 6.3: Calculating Probabilities of Events
Cherveny
Oct 3
Math 1004: Probability
Assigning Probability
Definition
A probability distribution assigns a probability to each outcome of
an experiment. It must satisfy two properties:
1. The probability of each outcome is between 0 and 1.
2. The sum of the probabilities of all outcomes is equal to 1.
Simple cases: A probability distribution is given by a chart of
outcomes and their assigned probabilities.
Cherveny
Oct 3
Math 1004: Probability
Assigning Probability
Finite Probability: Experiments have a finite number of outcomes,
which we denote by s1 , . . . , sN .
Definition
A probability distrubiton assigns a number P(si ) to each outcome
si such that
1. 0 ≤ P(si ) ≤ 1 for every i = 1, . . . , N.
2. P(s1 ) + P(s2 ) + · · · + P(sN ) = 1.
The number P(si ) is called the probability of si .
Cherveny
Oct 3
Math 1004: Probability
Rolling a Die
Example
Roll a six-sided die and observe the number rolled.
Outcome
1
2
3
4
5
6
Probability
1/6
1/6
1/6
1/6
1/6
1/6
Note: All probabilities are between 0 and 1 and
1 1 1 1 1 1
+ + + + + =1
6 6 6 6 6 6
so this is a probability distribution.
Cherveny
Oct 3
Math 1004: Probability
Letters of Missippi
Example
Choose a letter from the word MISSISSIPPI
Outcome
M
I
S
P
Probability
1/11
4/11
4/11
2/11
Note: All probabilities are between 0 and 1 and
1
4
4
2
+
+
+
= 1,
11 11 11 11
so this is a probability distribution.
Cherveny
Oct 3
Math 1004: Probability
Probability of Events
Recall an event of an experiment is a subset of the sample space.
Definition
The probability of an event E is the sum of the probabilities of all
the outcomes in E . In other words, if E = {x1 , . . . , xM }, then
P(E ) = P(x1 ) + · · · + P(xM )
Remark: ∅ is always an event. Its probability is P(∅) = 0.
Cherveny
Oct 3
Math 1004: Probability
Example
Example
A factory needs two raw materials. The probability of not having
enough of A is .05 and the probability of not having enough of B is
.03. A study determines the probability of a shortage of both A
and B is .01. What proportion of the time can the factory operate?
Answer: Let E be the event “there is a shortage of A” and F be
the event “there is a shortage of B”. We are given
P(E ) = .05
P(F ) = .03
P(E ∩ F ) = .01
A shortage of A or B is E ∪ F . By the inclusion-exclusion principle,
P(E ∪ F ) = P(E ) + P(F ) − P(E ∩ F ) = .05 + .03 − .01 = .07
The factory can operate with probability .93, or 93% of the time.
Cherveny
Oct 3
Math 1004: Probability
Odds vs Probability
Odds and probability are not the same thing!
Definition
The odds of an event is the ratio of the probability of the event to
the probability of the complement of the event:
Odds of E happening =
P(E )
P(E 0 )
Vegas Warning: In gambling, odds are often reported as “odds
against”. If the bookies say the Patriots odds are 3:7, they means
the Patriots don’t win 3 times for every 7 times they do win.
Cherveny
Oct 3
Math 1004: Probability
Odds Example
Example
Suppose the odds of winning a contest are .05. What is the
probability of winning this contest?
1
odds means you win the contest 1 time for
Answer: .05 = 20
every 20 times you lose. Therefore the probability of winning is
Cherveny
Oct 3
Math 1004: Probability
1
21 .
Practice
1. The probability that there will be a major earthquake in San
francisco during the next 30 years is .63. What are the odds
there won’t be a major earthquake?
2. The odds of rain today is “4 to 1”. What is the probability of
rain today?
3. Flip a fair coin three times and observe the sequence of
outcomes.
(a)
(b)
(c)
(d)
Make a probability distribution chart for this experiment.
What is the probability the second flip is heads?
What is the probability of all flips the same side?
What is the probability the first flip is heads and the last flip is
heads?
4. Flip a fair coin three times and observe the number of heads.
(a) Make a probability distribution chart for this experiment.
(b) What is the probability of one or two heads?
(c) What is the probability of one and two heads?
Cherveny
Oct 3
Math 1004: Probability
Practice Answers
1. Odds there won’t be a major earthquake =
.37
.63
= .587.
2. P(rain today) = 4/5 = .8
3. (a) There are 8 outcomes (HHH, HTH, etc), each with probability
1/8.
(b) 4/8
(c) P(same side) = P(HHH) + P(TTT) = 1/8 + 1/8 = 2/8
(d) 2/8
4. (a) The outcomes are 0, 1, 2, 3 with probabilities 18 , 38 , 83 , 18 ,
respectively.
(b) P(1 head or 2 heads) = P(1 head) + P(2 head) =
3/8 + 3/8 = 6/8
(c) 0
Cherveny
Oct 3
Math 1004: Probability