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Lecture 9 (September 27, 2011)
Probability Theory I
(
Flipping Coins for Computer Scientists)
Some Puzzles
Teams A and B are equally good
In any one game, each is equally likely to win
What is most likely length of a “best
of 7” series?
Flip coins until either 4 heads or 4 tails
Is this more likely to take 6 or 7 flips?
6 and 7 Are Equally Likely
To reach either one, after 5 games, it
must be 3 to 2
½ chance it ends 4 to 2; ½ chance it doesn’t
Teams A is now better than team B
The odds of A winning are 6:5
What is the chance that A will beat B
in the “best of 7” world series?
i.e., in any game, A wins with probability 6/11
Silver and Gold
A bag has two silver coins, another
has two gold coins, and the third has
one of each
One bag is selected at random.
One coin from it is selected at
random. It turns out to be gold
What is the probability that the
other coin is gold?
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Let us start simple…
A fair coin is tossed 100
times in a row
What is the probability that
we get exactly 50 heads?
The set of all outcomes is
{H,T}
100
There are 2
100
outcomes
Out of these, the number of
sequences with 50 heads is
100
50
/ 2
100
100
50
If we draw a random sequence, the
probability of seeing such a sequence:
= 0.07958923739…
The sample space
S
, the set of
all outcomes, is
{H,T}
100
The Language of Probability
“A fair coin is tossed 100
times in a row”
Each sequence in
S
is equally
likely, and hence has
probability
1/S=1/2
100
“What is the probability that we
get exactly 50 heads?”
Let E = {x in S x has 50 heads}
be the event
that
we see half heads.
The Language of Probability
Pr(E) = E/S = E/2
100
Pr(E) =
x in E
Pr(x) = E/2
100
Set S of all
2
100
sequences
{H,T}
100
Probability of event
E
= proportion of
E
in
S
Event E = Set of
sequences with
50
H
’s and
50
T
’s
100
50
/ 2
100
3
A fair coin is tossed 100
times in a row
What is the probability that
we get 50 heads in a row
?
The sample space
S
, the set of
all outcomes, is
{H,T}
100
formalizing this problem…
again, each sequence in
S
equally likely, and hence with
probability
1/S=1/2
100
Now E = {x in S x has 50 heads
in a row} is the event
of interest.
What is E?
2
50
50
HH
H
anything
HH
H
anything
T
2
49
HH
H
T
2
49
HH
H
T
2
49
HH
H
T
2
49
49
100
51
52 2
52
0
22
E
Total =
50 x 2
49
+2
50
=
52 x 2
49
If we roll a fair die, what is the probability
that the result is an even number?
½, obviously
True, but let’s take the trouble
to say this formally.
Each outcome x in
S
is equally likely, i.e.,
x in S,
the probability that x occurs is
1/6
.
sample space S = {1,2,3,4,5,6}
P( )
1
1
6
1
2
6
1
3
6
1
4
6
1
5
6
1
6
6
xx
2,4,6
E
1
1
1
3
1
P( )
6
6
6
6
2
E
Suppose that a dice is weighted so that
the numbers do not occur with equal
frequency.
E
2
1
3
4
2
P( )
6
12
12
6
3
E
P( )
1
1/ 6
2
2 / 6
3
1/12
4
5
6
3/12
table of frequencies
(proportions)
(probabilities)
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Language of Probability
The formal language of
probability is a crucial tool
in describing and analyzing
problems involving
probabilities…
and in avoiding errors,
ambiguities, and
fallacious reasoning.
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This note was uploaded on 11/03/2011 for the course CS 251 taught by Professor Gupta during the Spring '11 term at Carnegie Mellon.
 Spring '11
 Gupta

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