Introduction
Finite Probability
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Finite Probability
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Experiments and Outcomes
Experiment:
Tossing a coin
Outcomes:
{heads, tails}
Experiment:
Rolling a dice
Outcomes:
{1,2,3,4,5,6}
Experiment:
Rolling two dice
Outcomes:
{1,…,6}
×
{1,…,6}
or
{A
⊆
{1,…,6} :
|A|
≤
2}
Experiment:
Buying 3 lottery tickets (out of 100,000)
Outcomes:
3-element subsets of
{1,…,100000}
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Sample Space and Events
The set of all outcomes of an experiment is called the
sample space
Sometimes we are interested not in a single outcome, but an
event
that happens in several outcomes
Examples:
Get heads at least 3 times when tossing 5 coins
Win a prize in lottery
Get 2 aces in a poker hand
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Events
Let
S
be the sample space of a certain experiment.
An event is
any subset of
S
Examples:
Experiment:
Tossing 2 coins
Sample space:
S = {heads,tails}
×
{heads,tails}
Event:
Get exactly 1 heads
A = {(heads,tails),(tails,heads)}
Experiment:
Rolling 2 dice
Sample space:
S = {1,…,6}
×
{1,…,6}
Event:
The sum of the dice is
6
A = {(1,5),(2,4),(3,3),(4,2),(5,1)}
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Probability
In all our experiments each of the possible outcomes has the same
likelihood of occurrence, or the same probability of occurrence