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Math 011 – CHAPTER 4 Probability
DEFINITION
Event
Simple event
Sample space
Example:
Procedure
Example of Event
NOTATION:
Finding Probabilities:
1. Relative Frequency Approximation of Probability
2. Classical Approach to Probability (Requires equally likely outcomes)
3. Subjective Probability
Sample Space
Law of Large Numbers
Example: (Relative Frequency Probability)
A recent survey of 1010 adults in the U.S showed 202 of them smoke. Find the probability that
a randomly selected adult in the U.S. smokes.
Example: (Classical Approach)
Assuming that boys are equally likely, find the probability of getting children of the same
gender.
Example: (Subjective Probability)
a. Find the probability of our Statistics class being cancelled.
b. Find the probability that you’ll get stuck in the next elevator that you ride.
Rare Event Rule for Inferential Statistics:
Caution:
Example: Bills from 17 large cities in the U.S. were analyzed for presence of cocaine. Results: 23
of the bills were not tainted and 211 were tainted by cocaine. Based on results, find the
probability that a randomly selected bill is tainted by cocaine.
Example: If a year is randomly selected, find the probability that Thanksgiving Day in the U.S. is
on a) Wednesday; b) Thursday
Complement
202
Example: (Complement of a smoker) We found earlier that P(smoker) = 1010 = .20. Find the
probability of randomly selecting an adult in the U.S. who does not smoke.
Rounding off Probabilities
Example:
a.
b.
c.
d.
e.
Probability of 0.8208356
Probability of 0.0357469
Probability of 2/3
Probability of 4/8
Probability of 0.0002546
Interpreting Probabilities
An event is unlikely
An event has an unusually low number
An event has an unusually high number
Example:
a. A fair coin is tossed 1,000 times and exactly 500 heads result. Probability of getting 500
heads in 1000 tosses is 0.0252. Is this result unlikely? Is 500 heads unusually low or
unusually high?
b. A fair coin is tossed 1,000 times and 10 heads result. Is this result unlikely? Is 10 heads
unusually low or unusually high?
4-3 ADDITION RULE
Compound Event
Notation:
Example: Pre-Employment Drug Testing – Find the probability of selecting a subject who had a
negative test result or doesn’t use drugs.
Subject Uses Drugs
Subject Doesn’t Use Drugs
Formal addition Rule
Intuitive Addition Rule
Events A and B are disjoint
Example:
Disjoint event
Not disjoint events:
Positive Test Result
44
90
Negative Test result
6
860
Rule of Complementary Events
Example: According to a survey, 19.8% of college students take at least 1 online class. What’s
the probability of randomly selecting a college student who doesn’t take any online courses?
Example: Pre-employment Drug Testing – Find the probability of selecting a subject who had a
positive test result or uses drugs.
Subject Uses Drugs
Subject Doesn’t Use Drugs
Positive Test Result
44
90
Negative Test result
6
860
4-4 MULTIPLICATION RULE: BASICS
Notation:
Formal Multiplication Rule
Caution:
Two events A and B are independent
Two events are dependent
Example:
Subject Uses Drugs
Positive Test Result
44
Negative Test result
6
a. If two of the 50 subjects who used drugs are randomly selected with replacement, find
the probability that the first had a positive result and the second had a negative result.
b. If two of the 50 subjects who used drugs are randomly selected without replacement,
find the probability that the first had a positive result and the second had a negative
result.
Example: Among respondents asked which their favorite seat on a plane is, 492 said
window seat, 8 chose the middle seat, and 306 chose the aisle seat.
a. What is the probability of randomly selecting one of the people and getting one who
did not choose the middle seat?
b. If 2 are randomly selected without replacement, what is the probability that neither
chose the middle seat?
c. If 25 different people are randomly selected without replacement, what is the
probability that none chose the middle seat?
Example: (BIRTHDAYS) When 2 different people are randomly selected from your class, find the
indicated probability by assuming birthdays occur on the days of the week with equal
frequency.
a. Probability that two people are born on the same day of the week.
b. Probability that the two people are both born on a Monday.
4-5 MULTIPLICATION RULE Complements and Conditional Probability
Find the probability of getting at least one of some event:
Example: Find the probability of getting at least 7 when four digits are randomly selected with
replacement for a lottery ticket.
Conditional Probability
Example:
Subject Uses Drugs
Subject Doesn’t Use Drugs
Positive Test Result
44
90
Negative Test result
6
860
a. If one of the 1000 subjects is randomly selected, find the probability that the subject
had a positive result, given the subject actually uses drugs.
b. Find P(subject uses drugs | positive test results)
Confusion of the Inverse:
4-6 PERMUTATIONS AND COMBINATIONS Does order count?
DEFINTIONS
Permuatations
Combinations
Notation:
COUNTING RULES:
1. Fundamental Counting Rule
Example: How many different codes are there in a 2-character code consisting of a letter and
one digit?
In a 3-code with a letter and two digits?
2. Factorial Rule:
Example:
a. Find the number of ways that 3 different letters can be arranged?
b. Then find the number of arrangements for 5 different letters.
1. Permutations Rule (when all items are different)
Example: If 4 letters {a,b,c,d} are available and two are selected without replacement, find the
number of different permutations.
2. Permutations Rule ( when some items e the same as others)
Example: If 8 letters are available {a,a,b,b,c,c,c,c} and all are selected without replacement,
what is the number of different permutations?
3. Combination Rule
Example: If four letters {a,b,c,d} are available and two are selected without replacement, how
many different combinations are there?
Example: How many different ways can you touch three or more fingers to each other on one
hand?