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Probability

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Probability will allow inferences with a measure of reliability ... The student is male or lives in a coed dorm. The student is female and not a binge drinker ... – PowerPoint PPT presentation

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Title: Probability


1
Probability
  • 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

2
Events, 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

3
Events, 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

4
Events, 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

5
Events, 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
6
Events, Sample Spaces, and Probability
HH HT TH TT
1 2 3 4 6 5
7
Events, 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
8
Events, 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
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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

12
Problem 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.

13
Problem 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
14
Problem 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
15
Unions 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

16
Unions and Intersections
A
B
17
Complementary 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

18
Complementary 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

19
Complementary 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

20
Problem 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

21
Problem 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
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