Download 4-1 Notes (answers)

Notes 4.1 – What is Probability?
I. Def. Probability
+ A value between 0 and 1 that indicates the likelihood of an
event.
+ The closer to 1, the more likely the event.
+ written P(A); read “probability of event A”
II. 3 Methods of Assigning Probability
1) Intuition
2) Experimental Probability: obtained from a statistical experiment
with a predetermined # of trials.
f = frequency of an event
n
sample size
OR
f = # of times “it” happened
# of times you tried
3) Theoretical Probability: the way it SHOULD happen
(for equally likely outcomes)
P(A) = # of favorable outcomes
# of total outcomes
P(A) = # of ways “it” COULD happen
total # of POSSIBLE outcomes
Example: Darryl Dice rolled a 6-sided die 10 times and got a “5”
three times.
a) What is the experimental probability of rolling a 5? 3/10 = 0.3
b) What is the theoretical probability of rolling a 5?
1/6 = 0.1666
Law of Large Numbers – As sample size n increases, the relative
frequency of outcomes gets closer to the theoretical (actual)
probability value.
Example: Darryl Dice rolled his die 1000 times. What will happen to
his experimental probability?
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So which method do we use?: It depends on the information at hand
or that can be easily obtained
Examples
1. The director of the Readlot College Health Center wishes to open an eye
clinic. To justify the expense of such a clinic, the director reports the
probability that a student selected at random needs corrective lenses.
She took a random sample of 500 students to compute this probability and
found that 375 of them needed corrective lenses. What is the probability
that a student selected at random needs corrective lenses?
Answer: Since we are given a sample size can calculate experimental
probability.
P(student needs glasses) = 375/500 = 0.75
2. The Friends of the Library are hosting a fundraising barbecue. There are
four members of the cleanup committee, one of which is George. To see
who has to clean the grills, they are going to draw a name out of a hat.
Assuming each member has an equal chance of being selected, what is
the probability that George will have to clean the grills.
Answer: Since we know all the possibilities & there is no experimental data, we
should calculate theoretical probability.
P(George) = # favorable = 1 = 0.25
# total
4
3. Mr. Smith asks Joe what the probability of earning an A on the next test
will be. Based on past test grades, Joe is almost certain he will earn and
A. What specific number do you think Joe assigned the probability of
earning an A.
Answer: Because we aren’t given any actual values such as Joe’s past test
scores, the only way to assign a probability to this event is intuition.
Because Joe is almost certain he will earn an A, the probability should be close
to 1. We could say that P(Joe earning A) is between .9 and 1.
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III. Sample Space
Statistical experiment - any activity that results in a definite
outcome.
Sample Space – the set of all possible outcomes.
Example: Experiment = Tossing 1 coin  Sample Space = H T
Example
Today you have a 3 question true or false quiz. How many possible ways
could you answer the quiz?
(a) Finish listing the outcomes of the sample space.
TTT FTT TFT FFT
TTF FTF TFF FFF
(b) What is the probability that all three items will be false.
P(All F) = 1
8
only one way to get FFF
there are 8 possible outcomes
(c) What is the probability that exactly two items will be true?
There are 3 outcomes that have 2 true items: TTF, TFT, FTT
P(two T) = 3
8
IV. Complement of an Event
+ The sum of all the probabilities assigned to outcomes of a sample
space must be 1.
+ For an event A, the event not A is called the complement of A.
+ To compute the complement of A, we use the fact from above.
P( A) + P(not A) = 1
P(not A) = 1 – P(A)
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Example:
Event
Rolling a 3
Drawing a King
Will Rain
Pass
Prob&Stat
P(Event)
1/6
4/52 = 1/13
0.26
0.88
Complement
Not rolling a 3
Not drawing
king
Will not rain
Fail Prob&Stat
P(Complement)
5/6
12/13
0.74
0.12
V. Probability vs. Statistics
Probability and Statistics are like inverses.
Think about M&Ms in your cup…
You Know
Statistics
What you got out of the
cup (sample)
Probability
Exactly what is in the
cup
You Want to Know
What was in the cup to
begin with
What you will get out of
the cup
Created By: J & J Productions