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Probability
Introduction to Probability
Probabilities are associated with experiments where the outcome is not known in advance or
cannot be predicted
...
g
...

Probability measures and quantifies "how likely" an event, related to these types of experiment,
will happen

Probability is the quality or state of being probable
...


Sample Space
The sample space is the set of all possible outcomes in an experiment
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Example 3: If two dice are rolled, the sample space S is given by
S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }

Event
We define an event as some specific outcome of an experiment
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Example 4: A die is rolled (see example 1 above for the sample space)
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Event E is
given by
E = {2,4,6}

Example 5: Two coins are tossed (see example 2 above for the sample space)
...
Event E is given
by
E = {(HT),(TH)}

Example 6: Two dice are rolled (see example 3 above for the sample space)
...
Event E is given by
E = {(1,3),(2,2),(3,1)}
The value of a probability is a number between 0 and 1 inclusive
...
5

The occurrence of the event is just
as likely as it is unlikely

1
...
e
...

Let Ei denote the ith experimental outcome (elementary event) and P(Ei) is its probability of
occurring
...
For n
experimental outcomes:
𝑃(𝐸1 ) + 𝑃(𝐸2 ) +
...

𝑃(𝐸) =

𝑛(𝐸)
𝑛(𝑆)

The event of interest is "getting a 3"
...

The sample space S is given by S = {1,2,3,4,5,6}
...
so E = {2,4,6}, the even numbers on a die
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Identifying sample spaces and probabilities
Experiment 1: What is the probability of each outcome when a coin is tossed?

Outcomes: The outcomes of this experiment are head and tail
...
What is the
probability of landing on each color after spinning this spinner?
Sample Space:

{yellow, blue, green, red}

Probabilities:
P(yellow)

=

¼

P(blue)

=

¼

P(green)

=

¼

P(red)

=

¼

Experiment 3: What is the probability of each outcome when a single 6-sided die is rolled?
Sample Space:

{1, 2, 3, 4, 5, 6}

Probabilities:
P(1)

=

1/6

P(2)

=

1/6

P(3)

=

1/6

P(4)

=

1/6

P(5)

=

1/6

P(6)

=

1/6

Example 7: A die is rolled, find the probability of getting a 3
...
so E = {3}
...


Example 8: A die is rolled, find the probability of getting an even number
...
so E = {2,4,6}, the even numbers on a die
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Empirical Probability
It uses real data on present situations to determine how likely outcomes will occur in the future
...
We compute the Empirical probability when the
complete sample space is difficult to obtain

Independent Events
Independent Events: Two events may be independent when the actual happening of one
does not influence in any way the probability of the happening of the other one
...


Two events E and F are independent if and only if
𝑷(E and F) = 𝑷(𝑬)𝑷(𝑭) or 𝑷(𝑬 ∩ 𝑭) = 𝑷(𝑬)𝑷(𝑭)

Now, refer the above example and find the probability of obtaining two heads
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P(𝑭)

In words, probability of E and F is the probability of event E times the probability of event F
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𝑷(𝑮) …

Dependent Events
Two events are dependent if the outcome of one affects the outcome of the other
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The probability of the second card change after the first card is drawn
...
Tree diagrams are
particularly useful in probability since they record all possible outcomes in a clear and
uncomplicated manner
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Note: each toss of a coin is independent of the previous toss

Example on multiplication rule illustrated using a tree diagram for
dependent events
There are 2 blue and 3 red marbles in a bag
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Mutually Exclusive Events
Mutually Exclusive Events: Two events are known as mutually exclusive when the
occurrence of one of them excludes the occurrence of the other
...


Two events E and F are mutually exclusive if and only if
𝑷(𝑬 ∩ 𝑭) = 𝟎

Addition Rule for not mutually exclusive events
For any two events E and F,
P(E or F) = P(E) + P(F) – P(E and F)
E and F is the event consisting of simple events that belong to both E and F
...


Addition Rule for Mutually Exclusive Events
If events E and F have no simple events in common or cannot occur simultaneously, they are
said to be disjoint or mutually exclusive
...
The
̅ is all simple events in the sample space S that are not in the
complement of E is denoted by E
simple events E
...
6% of American households own
a dog
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316
P(E) = 1 − P(E)
= 1 − 0
...
684



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