Probability

1
Probability

• Introduction
• Experiments, Outcomes, Events

2
Introduction

• In this chapter, we discuss probability, which is
  the mathematics of chance.
• Many events in the world around us exhibit a
  random character, but by repeated observations of
  such events we can often determine long-term
  patterns (despite random, short-term
  fluctuations). Probability is the branch of
  mathematics devoted to the study of such events.

3
Probability of an Event

• The probability of an event is a number between 0
  and 1 that expresses the long-run likelihood that
  the event will occur.
• An event having probability .1 is rather unlikely
  to occur.
• An event with probability .9 is very likely to
  occur.
• An event with probability .5 is just as likely to
  occur as not.

4
Experiment, Trial, Outcome

• An experiment is an activity with an observable
  outcome.
• Each repetition of the experiment is called a
  trial.
• In each trial we observe the outcome of the
  experiment.

5
Example Experiment, Trial and Outcome

• Experiment 1 Flip a coin
• Trial One coin flip Outcome Heads
• Experiment 2 Allow a conditioned rat to run a
  maze containing three possible paths
• Trial One run Outcome Path 1
• Experiment 3 Tabulate the amount of rainfall in
  New York, NY in a year
• Trial One year Outcome 37.23 in

6
Sample Space

• The set of all possible outcomes of an
  experiment is called the sample space of the
  experiment. So each outcome is an element of the
  sample space.
• There are two types of sample spaces finite
  and infinite.
• Note The sample space of an event is
  equivalent to the universal set.

7
Example Sample Space

• An experiment consists of throwing two dice, one
  red and one green, and observing the numbers on
  the uppermost face on each. What is the sample
  space S of this experiment?
• Each outcome of the experiment can be regarded as
  an ordered pair of numbers, the first
  representing the number on the red die and the
  second the number on the green die.

8
Example Sample Space

• S (1,1), (1,2), (1,3), (1,4), (1,5),
  (1,6) (2,1), (2,2), (2,3), (2,4), (2,5),
  (2,6) (3,1), (3,2), (3,3), (3,4), (3,5),
  (3,6) (4,1), (4,2), (4,3), (4,4), (4,5),
  (4,6) (5,1), (5,2), (5,3), (5,4), (5,5),
  (5,6) (6,1), (6,2), (6,3), (6,4), (6,5),
  (6,6)

9
Event

• An event E is a subset of the sample space.
• We say that the event occurs when the outcome of
  the experiment is an element of E.

10
Example Event

• For the experiment of rolling two dice, describe
  the events
• E1 The sum of the numbers is greater
• than 9
• E2 The sum of the numbers is 7 or 11.
• E1 (4,6), (5,5), (5,6), (6,4), (6,5), (6,6)
• E2 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1),
• (5,6), (6,5)

11
Special Events

• Let S be the sample space of an experiment.
• The event corresponding to the empty set, Ø, is
  called the impossible event, since it can never
  occur.
• The event corresponding to the sample space
  itself, S, is called the certain event because
  the outcome must be in S.

12
Events as Sets

• Let E and F be two events of the sample space S.
• The event where either E or F or both occurs is
  designated by E ? F.
• The event where both E and F occurs is designated
  by E ? F.
• The event where E does not occur is designated by
  E '.

13
Example Events As Sets

• For the experiment of rolling two dice, let
• E1 The sum of the numbers is greater than 9
  and
• E3 The numbers on the two dice are equal.
• Determine the sets
• E1 ? E3
• E1 ? E3
• (E1 ? E3)'.

14
Mutually Exclusive Events

• Let E and F be events in a sample space S. Then E
  and F are mutually exclusive (or disjoint) if E ?
  F Ø
• If E and F are mutually exclusive, then E and F
  cannot simultaneously occur if E occurs, then F
  does not and if F occurs, then E does not.

15
Example Mutually Exclusive Events

• For the experiment of rolling two dice, which of
  the following events are mutually exclusive?
• E1 The sum of the dots is greater than 9
• E2 The sum of the dots is 7 or 11
• E3 The dots on the two dice are equal

16
Example

• A letter is selected at random from the word
  ALABAMA.
• a.) What is the sample space for this
• experiment?
• b.) Describe the event the letter chosen is a
• vowel as a subset of the sample space.

17
Example

• An experiment consists of tossing a coin four
  times an observing the sequence of heads and
  tails.
• a.) What is the sample space of this
  experiment?
• b.) Determine the event E1 more heads
• than tails occur.
• c.) Determine the event E2 the first toss is
  a tail.
• d.) Determine the event E1 ? E2.

18
Example

• Suppose that you observe the time (in minutes)
  that it takes a bank teller to deal with a
  customer.
• Describe the sample space.
• Is this a finite or infinite sample space?

19
Example

• Consider the following events
• A a person has a dog
• B a person is taking a math class
• C a person does not have any pets
• D a person is taking an English class
• Which, if any, of the events would be mutually
  exclusive? Explain.

20
Example The Game of Clue

• Anthony E. Pratt, then inventor of the game Clue,
  died in 1996. Clue is a board game in which
  players are given the opportunity to solve a
  murder that has six suspects, six possible
  weapons, and nine possible rooms, where the
  murder may have occurred.
• The six suspects are Colonel Mustard, Miss
  Scarlet, Professor Plum, Mrs. White, Mr. Green,
  and Mrs. Peacock.

21
Example The Game of Clue (continued)

• How could a sample space be formed with the
  entire solution to the murder, giving murderer,
  weapon, and site?
• How many outcomes would the sample space have?
• Let E be the event the murder occurred in the
  library. Let F be the event that the weapon was
  a gun. Describe E ? F and E ? F.