Download Probability - ESCCBUS271

Probability
Basic Concepts
Experiment
• An activity or measurement that results in an
outcome we cannot predict with absolute
certainty.
▫ A coin is flipped twice.
▫ We conduct a 10 year medical study on 12,000
participants
Event
• The simplest outcome of an experiment
• Toss two coins:
▫ 4 possible outcomes for each toss
▫ Each possible outcome is an event
▫ All the possible outcomes comprise the…
Sample space
• All the possible outcomes of an experiment:
▫
▫
▫
▫
(HH)
(HT)
(TH)
(TT)
• More examples
▫ One coin
▫ One die
▫ Red, white, and blue socks
 With and without replacement
Probability measure
• A function, P, from the subsets of the sample
space (Ω) that satisfies the following axioms:
1. P(Ω) = 1; that is, the sum of all the probabilities
in the sample space will = 1
2. P(A)≥0 for any event, A, that is included in the
sample space (Ω).
3. For mutually exclusive events A1 and A2 within
the same sample space
P(A1UA2) = P(A1) + P(A2)
Classical Probability
• Proportion of times an event can be expected to
occur
• All outcomes are equally likely:
Examples
• Flip a coin twice:
▫ P(HH) = P(HT) = P(TH) = P(TT) = 0.25
• Roll a die:
▫ P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Examples
• Consider events A and B (coins)
• We can determine their probabilities by adding
the probability of each occurrence.
▫ A: exactly one head [(HT), (TH)]
 P(A) = 0.25 + 0.25 = 0.50
▫ B: first flip head [(HH),(HT)]
 P(B) = ¼ + ¼ = ½
• If we have an experiment where we roll one die,
what is the probability of rolling less than a 3?
▫ Sample space
▫ Event
▫ P(A)
Compound Events
• Mutually Exclusive Events
▫ If one event occurs, the other cannot
• Exhaustive Events
▫ Set of events that include all the possible
outcomes of an experiment
▫ Events are exhaustive because one of them must
occur
▫ When events are mutually exclusive and
exhaustive, sum of their probabilities must equal 1
Intersection and Union
• Intersection of Events
▫ Two or more events occur at the same time in the
same experiment. [A and B, or A and B and C]
• Union of Events
▫ At least one of a number of possible events occurs
in the same experiment. [A or B, or A or B or C]
What’s the probability?
Experiment: the throw of one die
A: (observe an even number)
B: (observe a number <=3)
1. Describe A union B
2. Describe A intersection B
3. Calculate the probabilities of 1. and 2.
Example: Unions and Intersections
Age
Under 15
(B)
15 or older
(B’)
Male (A)
3477
5436
8913
Female (A’)
1249
1287
2536
4726
6723
11449
Addition Rules
• When events are mutually exclusive
• When events are not mutually exclusive
Practice
Region
Access to
Internet?
Northeast
Midwest
South
West
Yes
9.7
11.8
16.9
12.2
50.6
No
1.2
2.3
3.8
2.1
9.4
10.9
14.1
20.7
14.3
60.0
Identity 2 events that are mutually exclusive
Identify 2 events that intersect
Practice
Region
Access to
Internet?
Northeast
Midwest
South
West
Yes
9.7
11.8
16.9
12.2
50.6
No
1.2
2.3
3.8
2.1
9.4
10.9
14.1
20.7
14.3
60.0
What is the probability that a household would be in the South or Midwest or
have internet access?
What is the probability a household would be in the West and not have Internet
access?
A = 0.45
0.05
B = 0.15
What is the probability of AUB?
A
Bc?
• Marginal Probability
▫ Probability a given event will occur. No other
events are considered. P(A)
• Joint Probability
▫ Probability that two or more events will all occur.
P(A and B)
• Conditional Probability
▫ Probability that an event will occur given that
another event has already occurred. P(A|B)
Multiplication Rules
• Independent Events
▫ The occurrence of one has no effect on the
probability that the other will occur.
• Dependent Events
▫ The occurrence of one event influences the
probability of the other.
Multiplication Rules
• When events are independent
• When events are not independent
Fireworks Chart
Age
Under 15
(B)
15 or older
(B’)
Male (A)
3477
.304
5436
.475
8913
.779
Female (A’)
1249
.109
1287
.112
2536
.221
4726
.413
6723
.587
11449
1.000
Conditional Probability
Type of Policy (%)
Category
Fire
Auto
Other
Total %
Fraudulent
6
1
3
10
Nonfraudulent
14
29
47
90
Total
20
30
50
100
Conditional Probability Problem
• A corporation is going to select 2 of its regional
managers for promotion to VP. They have 6
male and 4 female regional managers. Assume
each manager has an equal probability (1/10) of
being selected.
▫ What is the probability that both people selected
for regional manager are male?
Practice Problems
• A fair coin is tossed 4 times. What is the
probability of getting at least one tail?
• What is the probability of getting exactly one
head?
• A card is drawn for a standard deck. What is the
probability that card will be a jack or a king?
More Practice Problems
• A standard pair of 6-sided dice is rolled. What is
the probability of rolling a sum greater than or
equal to 3?
• Three cards are drawn with replacement from a
standard deck. What is the probability that the
1st card will be a diamond, the 2nd card will be
black, and the third card will be a queen?
Still more practice problems
• 2 cards are drawn without replacement from a
standard deck. What is the probability of
choosing a club and then a black card?
• A box contains 6 green marbles and 19 white
marbles. What is the probability of choosing a
white marble if the first marble chosen was
white?
Counting
• Principle of multiplication
▫ m ways for event 1 to happen
▫ n ways for event 2 to happen
▫ Total number of possibilities = m x n
• If each of k independent events can happen n
different ways, the total number of possibilities
is nk
Counting
• Factorial rule of counting
▫ n! = n x (n-1) x (n-2) x … x 1
▫ 0! = 1
Counting
• Permutations: Number of possible arrangements
of n items in order
Counting
• Combinations: Order doesn’t matter. We
consider only the possible set of objects.
Counting Problems
• A 29-sided die is rolled 2 times. How many
different outcomes are possible?
• License plates consist of 2 letters followed by
three numbers. Duplicate digits are allowed.
How many different outcomes are possible?
• A doctor visits her patients during morning
rounds. In how many ways can she visit the 8
patients?
More Counting Problems
• A coordinator will select 6 songs from a list of 8
songs to compose a musical lineup. How many
different lineups are possible?
• 4 cards are chosen from a standard deck. How
many different 4 card hands are possible?
• A person tosses a coin 16 times. In how many
ways can he get 6 heads?
Practice
• The daily number in a state lottery is a 3-digit
integer between 000 and 999.
▫ What is the probability that the winning number
will be 555?
▫ Is the probability you found in part (a) an example
of classical, relative frequency, or subjective
probability?
▫ Today’s winning number is 347. You are going to
buy a ticket tomorrow and you plan to select
number 347. Is this a good idea? Why or why not?
Practice
Your company has two computer systems available for
processing telephone orders. Computer system A has a
10% chance of being down; computer system B has a 5%
chance of being down. The computer systems operate
independently. At least one system needs to work in order
to process new orders. For a typical telephone order,
determine the probability that:
▫ Neither computer system will be operational.
▫ Both computer systems will be operational.
▫ Exactly one of the computer systems will be
operational.
▫ What is the probability that the order can be
processed without delay?
Practice
• A security service employing 10 officers has been
asked to provide 3 officers for crowd control at a
local carnival. In how many different ways can
the firm staff this event?