6.2 Introduction to Probability
Terms:
“What are the chances of…?”
n  Personal probability (subjective)
¨  Based on feeling or opinion.
¨  Gut reaction.
n  Empirical probability (evidence based)
¨  Based on experience and observed data.
¨  Based on relative frequencies.
n  Theoretical probability
¨  Precise meaning.
¨  Based on assumptions.
¨  In the long run…
(formal)
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The Deal
n  Bag
o’ chips (poker chips).
¨ Some
are red.
¨ Some are white.
¨ Some are blue.
n  Action:
¨ We
Draw 1 chip from the bag.
don t know how many of each of the three
colors are in the bag.
¨ Assumption: Each chip has equal probability of
being chosen (Same size, same weight, same
texture, etc.)
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Possible outcomes:
n  Draw
a red chip.
n  Draw a
chip.
n  Draw a blue chip.
Definitions
Outcomes are the most basic possible results of
observations or experiments.
An easy way to list the possible outcomes: R, W, B
Example: Two-coin toss
n  Visual
of possible outcomes:
TT
TH
HT
HH
Note: each of these 4 outcomes is equally likely.
n  Suppose
we’re interested in the number
of heads…
Example: Two-coin toss
n  Visual
of possible outcomes:
TT
n  Number
TH
HT
HH
1
2
of heads…
0
1
Example: Two-coin toss
n  The
event of getting 2 heads.
n  Occurs
only when a HH is tossed.
n  One outcome coincides with this event.
n  The
event of getting 1 head.
n  Occurs
when either HT or TH is tossed.
n  Two outcomes coincide with this event.
n  Two outcomes represent the same event.
Definitions
An event is a collection of one or more outcomes that
share a property of interest.
Example: Two-coin toss
n  Probability
¨ P(1
head) = 2/4 = 0.50
n  Probability
¨ P(0
of getting 0 heads.
Sum to 1.
heads) = 1/4 = 0.25
n  Probability
¨ P(2
of getting 1 head.
of getting 2 head.
heads) = 1/4 = 0.25
We MUST get either 0, or 1, or 2 heads on the toss.
So, the probabilities of these 3 events MUST sum to 1.
Example: Two-coin toss
Number of heads:
2
1
1
0
Note: each of these 4 outcomes is equally
likely, and each has a ¼ chance of occurring.
Probability Rules
n  A probability
inclusive.
n  Something
is a number between 0 and 1,
0 ≤ P ≤1
has to happen rule.
¨ The
probability of the set of all possible
outcomes of a trial must be 1.
n  A
single coin toss: P(Head) + P(Tail) = 0.5 + 0.5 = 1
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Probability Rules
n  A probability
impossible.
of 0 means the the event is
n  P(event)=0.
n  A probability of
n  P(event)=1.
1 means the event is certain.
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Example: Roll of a 6-sided die
n  Six
possible outcomes: 1, 2, 3, 4, 5, 6
¨ The
event of rolling an odd number on a die.
n  Occurs
with a 1, 3, or 5.
n  Three outcomes in this event.
n  P(rolling
an odd) = 3/6 = ½ = 0.5
n  P(rolling a number larger than 6)=0
Note: each of these 6 outcomes is equally
likely, and each has a 1/6 chance of occurring.
Equally likely outcomes
Theoretical Method for Equally Likely Outcomes
Step 1. Count the total number of possible outcomes.
Step 2. Among all the possible outcomes, count the
number of ways the event of interest, A, can occur.
Step 3. Determine the probability, P(A), from
P(A) = number of ways A can occur
total number of outcomes
We used this technique for the events of
‘Rolling an odd’ and ‘Throwing 1 head’.
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