Announcements
Finite Probability
Wednesday, October 5th
I
MyMathLab 4 is due tonight!
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Problem Set 4 is due Friday Oct 7
Today: Sec. 6.3: Calculating Probabilities of Events
Use counting methods to find probabilities of events
when all outcomes are equally likely
Next Class: Sec. 6.3: Calculating Probabilities of Events II
Cherveny
Oct 5
Math 1004: Probability
Equally Likely Outcomes
If every outcome in a probability distribution has the same
probability, we say the outcomes are equally likely
Cherveny
Outcome
s1
s2
s3
..
.
Probability
1/N
1/N
1/N
..
.
sN
1/N
Oct 5
Math 1004: Probability
Equally Likely Outcomes
Theorem (Equally Likely Outcomes)
If all outcomes of an experiment are equally likely, then for any
event E
P(E ) =
# outcomes in E
# outcomes in the experiment
In other words, if there are M outcomes in event E and the sample
space has size N, then
P(E ) =
M
N
Goal: Ask yourself if all outcomes are equally likely. If so, then use
counting methods to find the probability of any event.
Cherveny
Oct 5
Math 1004: Probability
College Trustees
Example
Of the nine member of the board of trustees of a college, five
agree with the president on a certain issue. The president selects
three trustees at random and asks for their opinions. What is the
probability that at least two of them will agree with him?
Answer:
P(≥ 2 agree) = P(2 agree or 3 agree)
C (5, 2)C (4, 1) + C (5, 3)
=
C (9, 3)
≈ .595
Cherveny
Oct 5
Math 1004: Probability
Urn Example
Example
An urn contains five red balls and four white balls. A sample of
two balls is selected at random from the urn.
(a) Are the outcomes equally likely? How many are there?
(b) What is the probability that only red balls are selected?
(c) What is the probability that at least one white ball is selected?
Answer:
(a) Yes; C (9, 2)
(b) P(only red) =
C (5,2)
C (9,2)
≈ .278
(c) P(≥ 1 white) = 1 − P(0 whites) = 1 −
Cherveny
Oct 5
C (5,2)
C (9,2)
Math 1004: Probability
≈ .722
Practice
1. An urn contains seven orange balls and five black balls. A
sample of three balls is selected at random from the urn. Find
the probability that
(a) only orange balls are selected
(b) at least one black ball is selected
2. Of the 15 members on a Senate committee, 10 plan to vote
“yes” and 5 plan to vote “no” on an important issue. A
reporter attempts to predict the outcome of the vote by
questioning six of the senators. Find the probability that this
sample is exactly representative of the final vote.
3. A classroom contains 22 children (12 boys and 10 girls).
Seven students are chosen to lineup on the blackboard.
(a) What is the probability that no boys are chosen?
(b) What is the probability that the first three in line are boys?
(c) What is the probability that no two boys and no two girls are
standing next to each other?
(d) What is the probability that the third child in line is a girl?
Cherveny
Oct 5
Math 1004: Probability
Practice Answers
1. An urn contains seven orange balls and five black balls. A
sample of three balls is selected at random from the urn. Find
the probability that
(a)
(b)
C (7,3)
C (12,3)
(7,3)
1 − CC(12,3)
2. Need 4 of the 6 to say yes. P(4 yes) =
C (10,4)·C (5,2)
C (15,6)
3. A classroom contains 22 children (12 boys and 10 girls) in
which seven students are chosen to go to the blackboard.
(a)
P(10,7)
P(22,7) (which is same
10·9·8·19·18·17·16
(which
P(22,7)
as
C (10,7)
C (22,7) )
(b)
is same as CP(10,3)
(22,3) )
10·12·9·11·8·10·7+12·10·11·9·10·8·9
(c) P(GBGBGBG or BGBGBGB) =
P(22,7)
(d) 10
22
Cherveny
Oct 5
Math 1004: Probability