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6.2 Introduction to Probability
part 2
! From Tables to Probability
" Computing relative frequency probabilities from a
table.
! Probability Distributions
! The table below provides information on the
hair color and eye color of 592 statistics
students (The American Statistician, 1974).
Summary Table
Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592 3
! If a student is drawn at random from this
population…
" What is the probability that they are blond?
! ANS: 127/592 = 0.
" What is the probability that they are NOT blond?
! ANS: P(NOT blond) = 1-P(blond) =1-127/592 = 0.
! ANS: (71 +286 + 108)/592 = 465/592 = 0.
Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592
! If a student is drawn at random from this
population…
" What is the probability that they are blond
and have brown eyes?
! ANS: 7/592 = 0.
" What is the probability that they have red
hair and have green eyes?
! ANS: 14/592 = 0.
Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592 5
! Based on this table…
" What is the probability that a blond student
has brown eyes?
! ANS: 7/127 = 0.
" What is the probability that a brown-haired
student has hazel eyes?
! ANS: 54/286 = 0.
Eye Color Row Brown Blue Hazel Green Total Black 68 20 15 5 108 Hair Brown 119 84 54 29 286 Color Red 26 17 14 14 71 Blond 7 94 10 16 127 592
We’ve subsetted to a smaller
population from which we are
drawing a person
Relative Frequency Method
! Step 1. Repeat or observe a process
many times and count the number of times
the event of interest, A , occurs.
! Step 2. Estimate P ( A ) by
! P ( A ) =
number of times A occurred
total number of observations
! A probability distribution shows the possible
values that could occur, and the probability
for each occurring.
Probability Distributions
The probability of each value is
shown by the height of the bar.
! Find the probability distribution for the ‘sum
of two dice’.
Example: Roll two 6-sided dice
Probability (^2 3) Sum of two dice 4 5 6 7 8 9 10 11 12
Most likely sum
is a sum of 7.
Only one way
to get sum of
12 (double 6’s).
P(sum of 12) =
! Find the probability distribution for the ‘sum
of two dice’.
Example: Roll two 6-sided dice
Probability (^2 3) Sum of two dice 4 5 6 7 8 9 10 11 12
Recall,
P(sum of 11) =
And that is the
height of the bar
above the number
! Find the probability distribution for ‘the
number that turns up’.
Example: Roll one 6-sided dice
Each of 6 values is
equally likely, at 1/
value on die probability 1 2 3 4 5 6
0.^ 0.^ 0.^ 0.^ 0.^
This is a uniform distribution.
! How to make a probability distribution.
1) List all possible outcomes. Use a table or
figure it is helpful.
2) Identify outcomes that represent the same
event. Find the probability of each event.
3) Make a table in which one column lists
each event and another column lists each
probability. The sum of all the probabilities
must be 1.
Probability Distributions
! Random experiment: Toss Three Coins
! Make a probability distribution for the
number of heads that occurs.
1) List the possible outcomes (equally likely):
HHH,HHT,HTH,THH,HTT,THT,TTH,TTT
2) Identify outcomes that represent the same
event. There are four possible events:
0, 1, 2, or 3 heads.
Example: Making a probability
Probability Distribution