Title: Probability
1Probability
- Want to be able to make inferences about a
population from a sample or samples - Probability will allow inferences with a measure
of reliability for the inferences - Initially, we will assume that the population is
known and will calculate the probability of
observing various samples from the population
i.e., use the population to infer the probable
nature of the sample
2Events, Sample Spaces, and Probability
- Experiment---an act or process of observation
that leads to a single outcome that cannot be
predicted with certainty - Observing the up face when tossing a coin
- Observing the up faces on the toss of 2 coins
- Observing the up face on a die toss
3Events, Sample Spaces, and Probability
- Sample point---most basic outcome of an
experiment - Observe H or T
- Observe HH or HT or TH or TT
- Observe 1 or 2 or 3 or 4 or 5 or 6
4Events, Sample Spaces, and Probability
- Sample space---collection of all sample points
for the experiment - S H,T
- S HH, HT, TH, TT
- S 1, 2, 3, 4, 5, 6
5Events, Sample Spaces, and Probability
The probability of a sample point is a number
between 0 and 1 that measures the likelihood that
the outcome will occur when the experiment is
performed The sum of the probabilities of all
sample points in a sample space is 1
H T
HH HT TH TT
6Events, Sample Spaces, and Probability
HH HT TH TT
1 2 3 4 6 5
7Events, Sample Spaces, and Probability
Event---A specific collection of sample
points The probability of an event is calculated
by summing the probabilities of the sample points
in the sample space for the given event
8Events, Sample Spaces, and Probability
- Problem 3.6, page 117
- Two fair dice are tossed and the up face on each
die is recorded - What constitutes a sample point and what is the
sample space? - Find the probability of observing each of the
following events - A A 3 appears on each of the two dice
- B The sum of the numbers is even
- C The sum of the numbers is equal to 7
- D A 5 appears on at least on of the dice
- E The sum of the numbers is 10 or more
9(No Transcript)
10- Assuming a pair of fair dice, each of the 36
outcomes are equally likely - A A 3 appears on each of the two dice
- A(3,3)? P(A) 1/36
- B The sum of the numbers is even
- B(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),
- (3,3),(3,5),(4,2),(4,4),(4,6),(5,1
),(5,3), - (5,5),(6,2),(6,4),(6,6)
- ? P(B)
18/361/2 -
11- C The sum of the numbers is equal to 7
- C (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)
- ? P(C) 6/36
1/6 - D A 5 appears on at least on of the dice
- D (1,5),(2,5),(3,5),(4,5),(5,5),(6,5),
- (5,1),(5,2),(5,3),(5,4),(5,6)
- ? P(D)
11/36 - E The sum of the numbers is 10 or more
- E (4,6),(5,5),(5,6),(6,4),(6,5),(6,6)
- ? P(E) 6/36
1/6
12Problem 3.21, page 119
- Three people play a game called Odd Man Out.
Each player flips a coin until the outcome (H or
T) for one of the players is not the same as the
other two players. Find the probability that the
game ends after only one toss by each player. - Translates to find the probability that either
exactly one of the coins will be H or exactly one
of the coins will be T.
13Problem 3.21, page 119
- List possible outcomes from 3 tosses
-
Lose if exactly one H or exactly one T therefore
lose if get HHT, HTH, THH, HTT, THT, or TTH Each
event is equally likely with P 1/8 P(Exactly 1
T or exactly 1 H) P(HHT) P(HTH) P(THH)
P(HTT) P(THT) P(TTH) 1/81/81/81/81/8
1/8 6/8 3/4
HHH HHT HTH THH HTT THT TTH TTT
14Problem 3.21, page 119
- Now assume that one of the players uses a
2-headed coin hoping to reduce his chances of
being the odd man out. Will this ploy be
successful? - Assume that the 1st player uses the 2-headed
coin. - The new sample space is HHH, HHT, HTH, HTT
-
Each event is equally likely with P 1/4 The
probability that the 1st player is the odd man
out is P(HTT) 1/4 From the 1st part, the
probability that the 1st player is the odd man
out is P(THH) P(HTT) 1/8 1/8 1/4
15Unions and Intersections
- Compound events---defined as a composition of two
or more other events - They can be formed in two ways
- Union---the union of two events A and B, denoted
as , is the event that occurs if either
A or B or both occur on a single performance of
an experiment - Intersection---the intersection of two events A
and B, denoted as , is the event that
occurs if both A and B occur on a single
performance of the experiment
16Unions and Intersections
A
B
17Complementary Events
- Complement---the complement of event A, denoted
as , is the event that A does not occur the
event consisting of all sample points not in A - May be useful for events with large numbers of
possible outcomes
18Complementary Events
- Example
- Toss a coin ten times and record the up face
after each toss. What is the probability of
event A Observe at least one head - The list of possible sample points is long
- HHHHHHHHHH, HHHHHHHHHT,
- HHHHHHHHTH, HHHHHHHTHH, etc.
- There are possible outcomes
19Complementary Events
- Example
- Likewise, the list of sample points in event A is
also long---all sequences that contain at least
one H. Consider the complement of A - No heads are observed in 10 tosses
- TTTTTTTTTT
- and P( ) 1/1,024
- Now, P(A) 1 P( ) 1 1/1,024
1,023/1,024 -
0.999
20Problem 3.32, page 129
- A study of binge alcohol drinking by college
students was published by the Amer. Journal of
Public Health in July 95. Suppose an experiment
consists of randomly selecting one of the
undergraduate students who participated in the
study. Consider the following events - A The student is a binge drinker
- B The student is a male
- C The student lives in a coed dorm
21Problem 3.32, page 129
- Describe each of the following events in terms of
unions, intersections and complements - The student is male and a binge drinker
- The student is not a binge drinker
- The student is male or lives in a coed dorm
- The student is female and not a binge drinker