Download Probability

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Probability
Probability is a mathematical measure of the likelihood of an event occurring in any one trial
or experiment carried out in prescribed conditions.
Probability Experiments
A probability experiment is any process whose result is determined by chance.
Examples:
 Roll a die. Flip a Coin.
 Draw a card from a shuffled deck
 Choose a name from a hat.
Outcome – Any possible result of a probability experiment.
Sample Space – The collection (set) of all possible outcomes for an experiment. The sample
space is often denoted by S.
Examples:
 The sample space of the experiment Flip a Coin has 2 outcomes heads and tails, so we
could write:
S  {heads, tails}
 The experiment of rolling a die has 6 possible outcomes 1-6, so:
S  {1, 2,3, 4,5,6}
 Measuring the blood pressure of a person at random.
S  0  x  200
Note: The first two examples represent a DISCRETE sample space (consisting of finite or
countably infinite set of outcomes) while the third one is an example of a CONTINUOUS sample
space (consists of a finite or infinite interval of real numbers).
Event – Any collection of outcomes from the sample space or a subset of the sample space. We
will generally use capital letters to represent events.
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Example 1: Rolling an even number is an event for the experiment of rolling a die. If we call
E  {2, 4,6}
this event E, we could write:
Example 2: A coin is tossed twice. List the elements of the sample space S, and list the
elements of the event consisting of at least one head.
A probability function is a function P that assigns to each event E, a number P( E ) called the
probability of that event, such that the following three hold:
i.
ii.
iii.
0  P( E )  1 .
P( S )  1 .
If A and B are mutually exclusive events, then P( A  B)  P( A)  P( B)
An event E is called impossible if P( E )    0 .
An event E is called certain if P( E )  S  1 .
Calculating the Probability of an Event
P( E ) can be calculated in two different ways:
Empirical Probability – A probability calculated based on previous known results. The relative
frequency of the number of times the event has previously occurred is taken as the indication
of likely occurrences in the future.
Examples:
 We could calculate the probability of the event of obtaining a sum of 7 when two dice
are rolled by rolling 2 dice many times (say 1000) and calculating what percent of the
time a 7 was rolled.
 We can calculate the probability that a basketball player will make a free throw by
calculating the percentage of the time that the player has made free throws in the past.
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If an experiment is performed n times, and we count the number of times that event E actually
occurs, then based on these actual results,
P( E ) 
frequency of E
n
Example: A random batch of 240 components is subjected to strict inspection and 20 items are
found to be defective. Therefore if we pick a component at random from this sample, the
chance of it being faulty is
20
1
 .
240 12
So, if A  faulty component
Then P( A) 
1
12
Therefore a run of 600 components from the same machine would be likely to contain
1
 600  50 defectives.
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Exercises:
 A player has made 527 free throws in 698 attempts. Calculate the probability that the
player makes a free throw on the next attempt.
 In 2010 there were 62,318 flights departing from Jomo Kenyatta International Airport.
Of these, 8,821 were late in departing. Calculate the probability that a flight departing
from JKIA will be delayed.
Given a frequency table for a set of data values, we can calculate the probability that a data
value falls into any data class.
Example: Consider the following frequency table of exam scores:
Class
90-99
80-89
70-79
60-69
50-59
40-49
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Frequency ( f )
4
6
4
3
2
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If event A is the event that a student scores between 90 and 99 on the exam, then:
P( A) 
number of students scoring 90-99 4

 0.20
total number of students
20
Notice that P( A) is just the relative frequency of the 90-99 data class.
Exercises:
 Let B be the event that a student scores between 80 and 89 on the exam. Calculate
P( B) .
 Let E be the event that a student passes the exam (with a grade of C or higher).
Calculate P( E )
Classical Probability (Theoretical Probability) – A probability calculated for an experiment in
which each outcome is equally likely.
The probability of an event in such an experiment can be calculated by determining the
fraction (or percentage) of outcomes that are in the event.
P( E ) 
number of ways in which event E can occur
total number of outcomes in S
Examples:
 When flipping a fair coin, the two possible outcomes heads and tails are equally likely,
so:
P(heads)  P(tails) 
1
2
 When rolling a fair die each possible outcome from 1 to 6 is equally likely, with
probability
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1
.
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Exercise:
A statistics class contains 14 males and 20 females. A student is to be selected by chance and
the gender of the student recorded.
(a) Give a sample space S for the experiment.
(b) Is each outcome equally likely? Explain.
(c) Assign probabilities to each outcome.
Complement of an Event
The complement of an event E is denoted by E c , and is the event that contains every outcome
for the experiment except for those in E. It is essentially the event that E does not occur.
Example 1: If E is the event that a 5 is showing when a die is rolled, then E c is the event that a
c
5 is not, so E c contains 5 outcomes: E  {1, 2,3, 4,6}
Since every event either occurs or does not occur, the sum of the probabilities of an event and
its complement is always 1. This fact can be used to find the probability of the complement of
an event as follows:
P ( E c )  1  P( E )
Example 2: If the probability that a flight will be delayed is 0.13, then the probability that it will
not be delayed must be 1  0.13  0.87 .
Often it is easier to calculate the probability of the complement of an event than the
c
probability of the event itself. We can then use the fact that P( E )  1  P( E ) to find the
probability of the original event E.
One common example is when we want to find the probability of at least one occurrence of an
event in some number of trials.
Example 3: If a die is rolled 4 times, find the probability that at least one of the rolls is a six.
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Conditional Probability
A conditional probability for an event is calculated when some additional information about
the outcome of the experiment is known.
Suppose A and B are two events for an experiment, and suppose that it is known that event B
has occurred. The probability that A occurs given that B has already occurred is denoted by
P( A | B) and is called the conditional probability of A given B.
Example 1:
Consider the experiment of drawing two balls from a bag which has one red ball, one green
and one blue ball without replacement. Let event A be the event that the first ball is red, and
event B be that the second ball is green. We can calculate that:
P( A) 
1
3
P( B | A) 
1
2
Example 2: A die is rolled; find the probability of getting a 4 if it is known that an even number
occurred when the die was rolled.
Solution:
If it is known that an even number has occurred, the sample space is reduced to 2, 4, or 6.
Hence the probability of getting a 4 is
1
since there is one chance in three of getting a 4 if it is
3
known that the result was an even number.
Example 3: Two coins are tossed. Find the probability of getting two tails if it is known that one
of the coins is a tail.
Example 4:
A box contains 100 copper plugs, 27 oversize and 16 undersize. A plug is taken, tested but not
replaced: a second plug is then treated similarly. Determine the probability that
a) Both plugs are acceptable,
b) The first is oversize and second undersize
c) One is oversize and the other undersize.
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Independence and Dependence
If the occurrence or non-occurrence of event A does not affect the chance of event B
happening and vice-versa, then
P( A | B)  P( A) and P( B | A)  P( B)
and A and B are called independent events. If the above equations do not hold, then the two
events are called dependent.
Examples:
 In rolling a die twice, the outcome of the first throw will not affect the probability of
throwing a six on the second throw. The two events are independent.
 Consider the experiment of drawing two balls, one at a time, from a bag of four red balls
4
. If the ball
7
4
is replaced, then the probability of drawing a red ball on the second occasion is still .
7
and three green balls. The probability that the first ball drawn will be red is
(Independent events).
However if the first red ball wasn’t replaced, then the probability of drawing a red the
second time, is now
3 1
 (dependent events).
6 2
Note: Two events are dependent only if one event’s occurrence affects the likelihood of the
other. So events from two different trials of an experiment are always independent.
Combining Events Using AND
Given two events A and B we define the event A and B to be the event that events A and B
both occur at the same time. We find the probability of the event A and B using the
Multiplication Rule.
The Multiplication Rule – If A and B are two events for an experiment, then:
P( A and B)  P( A)  P( B | A)  P  B   P( A | B)
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Equivalently
P( A  B)  P( A)  P( B | A)
The formula is often used when the events occur in sequence so that event A is followed by
event B.
Example1: The probability that an automobile battery subject to high engine compartment
temperature suffers low charging current is 0.7. The probability that a battery is subject to high
engine compartment temperature is 0.05.
Let C denote the event that a battery suffers low charging current, and let T denote the
event that a battery is subject to high engine compartment temperature. The probability that a
battery is subject to low charging current and high engine compartment temperature is
P(C and T )  P(T )  P(C | T )  0.05  0.7  0.035
If A and B are independent events, then P( B | A)  P( B) so that the formula for calculating the
probability of A and B becomes:
P( A and B)  P( A)  P( B)
Example2: If two dice are rolled, the probability of rolling double 6 can be found using the
above formula. The probability of getting a 6 on each die is 1 and the events of each roll are
6
independent from each other. Thus the probability of a “double 6” is just:
1 1 1
P(6 on die #1 and 6 on die #2)   
6 6 36
This rule can also be extended to 3 or more events. For example the probability of rolling a 6
on each of 4 rolls would be:
1 1 1 1 1
1
P(four sixes)      4 
6 6 6 6 6
1296
Exercise:
 The probability of a flight departing from Moi International Airport being delayed is
0.14. Find the probability that 3 randomly chosen flights are all delayed.
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 There are 2000 voters in a town. Consider the experiment of randomly selecting a voter
to be interviewed. The event A consists of being in favor of more stringent building
codes; the event B consists of having lived in the town less than 10 years. The following
table gives the numbers of voters in various categories.
A
Favor more
stringent codes
B
Less than
10 years
Ac
Do not favor more
stringent codes
100
700
1000
200
Bc
At least
10 years
Find each of the following:
(a) P(A)
(b) P( B c )
(c) P(A and B)
Note: From the multiplication rule, we get the formula for the conditional probability of two
events A and B:
P( A | B) 
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P( A and B)
P  B
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Combining Events Using OR
Given two events A and B we define the event A or B to be the event that at least one of the
events A or B occurs. We can find the probability of the event A or B using the Addition Rule.
The Addition Rule – If A and B are two events for an experiment, then:
P( A or B)  P( A)  P( B)  P( A and B)
Equivalently
P( A  B)  P( A)  P( B)  P( A  B)
Two events are called mutually exclusive if both events cannot occur at the same time. In this
case P( A and B)  0 , so the Addition Rule simplifies to:
P( A or B)  P( A)  P( B)
Example 1: If two dice are rolled, then the event A of rolling a 7 and the event B of rolling an
11 are mutually exclusive, so that the probability of rolling 7 or 11 is:
Example 2: Roll two dice, red and blue. What is the probability of landing either a red six or a
blue six?
Exercises:
 If two dice are rolled, find the probability that the number on the first die is even or the
number on the second die is 1.
 A package of candy contains 8 red pieces, 6 white pieces, 2 blue pieces, and 4 green
pieces. If a piece is selected at random, find the probability that it is
a. White or green.
b. Blue or red
 In a psychology class, there are 15 seniors and 18 juniors. Six of the seniors are males
and 10 of the juiors are males. If a student is selected at random, find the probability
that the student is
a. A junior or a male.
b. A senior or a female.
c. A junior.
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Discrete Probability Spaces
In a discrete sample space, S, the probability of an event E, denoted as P  E  , equals the sum
of the probabilities of the outcomes in E.
Example 1: A random experiment can result in one of the outcomes a, b, c, d  with
probabilities 0.1, 0.3, 0.5, and 0.1 respectively. Let A  a, b , B  b, c, d  , C  d  .
i) P  A , P  B  , P  C 
Find:
ii) P  Ac 
iii) P  A  B  , P  A  B  , P  A  C 
1
Example 2: Let S  a1 , a2 , a3 , a4  and let P be a probability defined on S. Given P  a3   and
6
P  a4  
i)
1
,find
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P  a1  if P  a2  
1
3
ii) P  a1  and P  a2  if P  a1   2P  a2  .
Exercise:
Let A and B be events in S such that P  A  B   , P  A  B   , P  Ac   . Find:
i) P  B 
7
8
1
4
5
8
ii) P  Ac  Bc 
iii) P  Ac  Bc 
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