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Chapter 5: Probability
Chapter 6: Probability
Distributions
Read Chapter 6
Probability
The probability of an event is defined as its relative
frequency.
Relative Frequency Definition
That is, for an event A,
P(A)= number of ways A happens
total number of outcomes
Empirical/mechanistic approach to probability
American or European Roulette?
A common application of expected value is in gambling. For example,
an
American Roulette
wheel has 38 equally likely outcomes. A
winning bet placed on a single number pays 35to1 (this means that
you are paid 35 times your bet and your bet is returned, so you get
36 times your bet). So considering all 38 possible outcomes, the
expected profit resulting from a $1 bet on a single number is:
−
$1(37/38) + $35(1/38) =
−
$0.0526.
Therefore one expects, on average, to lose over five cents for every
dollar bet, and the
expected value
of a one dollar bet is $0.9474.
On an European Roulette table:
−
$1(36/37) + $35(1/37) =
−
$0.0270.
The
expected value
of a one dollar bet is now $0.9730.
Properties of Probability
For every event A in
Ω
(the set of all
events),
0
≤
P(A)
≤
1
P(
Ω
)=1
The set of outcomes that are not in event
A is called the complement of A, A
c
.
P(A) = 1 – P(A
c
)
Properties of Probability
For two events A and B
P(AUB) = P(A) + P(B) – P(A
∩
B).
http://wwwgroups.dcs.standrews.ac.uk/~history/Mathematicians/Venn.html
Example:
If a single card is randomly drawn
from an ordinary deck of 52 cards, find the
probability that it will be a club or a face card.
P(E
∪
F) = P(E) + P(F) – P(E
∩
F)
P(club
∪
face) = P(club) + P(face) – P(club
∩
face)
P(club
∪
face) =
13
+
12

3
=
22
=
11 .
52
52
52
52
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