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Слайды и текст к этой презентации:
№5 слайд
Содержание слайда: Probability
Let’s Make a Deal
The famous game show (you might have seen this problem in your books)
Participant is given a choice of three doors. Behind one door is a car, behind the others, useless stuff. The participant picks a door (say door 1). The host, who knows what is behind the doors, opens another door (say door 3) which has the useless stuff. He then asks the participant if he would like to switch (pick door 2)?
Is it to participant’s advantage to switch or not?
№7 слайд
Содержание слайда: Probability
Precise Description
The car is equally likely to be hidden behind the three doors.
The player is equally likely to pick each of the doors.
After the player picks a door, the host must open a different door (with the useless thing behind it) and offer the player to switch.
When a host has a choice of which door to pick, he is equally likely to pick each of them.
Now here comes the question:
“What is the probability that a player who switches wins the car?”
№12 слайд
Содержание слайда: Probability
Solving Problems Involving Probability
Step 1: Finding the sample space
Set of all possible outcomes of a random process
To find this, we must understand the quantities involve in the random process
Quantities in the above problem:
The door concealing the car
The door initially chosen by the player
The door that host opens to reveal the useless thing
№14 слайд
Содержание слайда: Probability
Finding the Sample Space
Every possible value of these quantities is called an outcome.
And (as said earlier) the set of all possible outcomes is called the sample space
A tree structure (Possibility tree) is a useful tool for keeping track of all outcomes
When the number of possible outcomes is not too large
№18 слайд
Содержание слайда: Probability
Possibility Tree --- Cont.
The car can be at any of these three locations
For each possible location of the car, the player can choose any of the three doors
Then the final possibility is regarding the host opening a door to reveal the useless thing
Overall tree turns out to be
№25 слайд
Содержание слайда: Probability
Solving Problems Involving Probability
Step 2: Defining the Events of Interest:
Remember, we want to answer the questions of type:
“What is the probability that … ?”
Replacing the “…” with some specific event. For example,
“What is the probability that the car is behind door C?”
Doing this reduces S to some specific outcomes, called event of interest.
№42 слайд
Содержание слайда: Set Theory and Probability
Sample Space S : A nonempty countable set.
An element is called an outcome.
A subset of S is called an event to which a probability is assigned.
If you look closely, you will realize that to calculate this probability we first have to count the elements in these sets.
№44 слайд
Содержание слайда: Probability
The Addition Rule
The basic rule underlying the calculation of the number of elements in a union or difference or intersection is the addition rule.
This rule states that the number of elements in a union of mutually disjoint finite sets equals the sum of the number of elements in each of the component sets.
Theorem 9.3.1:
Suppose a finite set A equals the union of k distinct mutually disjoint subsets A1, A2, …., Ak. Then
N(A) = N(A1)+N(A2)+…+ N(Ak)
№45 слайд
Содержание слайда: Probability
The Addition Rule---Cont.
Example: A computer access password consists of from one to three letters chosen from the 26 in the alphabet with repetitions allowed. How many different passwords are possible?
Solution: The set of all passwords can be partitioned into subsets consisting of those of length 1, those of length 2, and those of length 3 as shown in the figure below.
№46 слайд
Содержание слайда: Probability
The Addition Rule---Cont.
By the addition rule, the total number of passwords equals
the number of passwords of length 1, plus the number of
passwords of length 2, plus the number of passwords of
length 3.
Now the,
Number of passwords of length 1= 26
Number of passwords of length 2 =262
№48 слайд
Содержание слайда: Probability
The Difference Rule
An important consequence of the addition rule is the fact that if the number of elements in a set A and the number in a subset B of A are both known, then the number of elements that are in A and not in B can be computed.
Theorem 9.3.2: The Difference Rule:
If A is finite set and B is a subset of A, then
N(A-B) = N(A) – N(B)
№50 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
The difference rule holds for the following reason: If B is a subset of A, then the two sets B and A – B have no elements in common and B (A – B) = A. Hence, by the addition rule,
N(B) + N(A – B) = N(A).
Subtracting N(B) from both sides gives the equation
N(A – B) = N(A) – N(B).
№51 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
Example:
A typical PIN (personal identification number) is a sequence of any four symbols chosen from the 26 letters in the alphabet and the ten digits, with repetition allowed.
a. How many PINs contain repeated symbols?
b. If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol?
№54 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
Example --- Cont.:
There are 364 = 1,679,616 PINs when repetition is allowed,
and there are 36 35 34 33 = 1,413,720
PINs when repetition is not allowed.
Thus, by the difference rule, there are
1,679,616 – 1,413,720 = 265,896
PINs that contain at least one repeated symbol.
№56 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
Example --- Cont.:
There are 1,679,616 PINs in all, and by part (a) 265,896 of these contain at least one repeated symbol.
Thus, by the equally likely probability formula, the probability that a randomly chosen PIN contains a repeated
symbol is
№58 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
An alternative solution to Example 3(b) is based on the observation that if S is the set of all PINs and A is the set of all PINs with no repeated symbol, then S – A is the set of all PINs with at least one repeated symbol.
It follows that
№59 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
An alternative solution to Example 3(b) is based on the observation that if S is the set of all PINs and A is the set of all PINs with no repeated symbol, then S – A is the set of all PINs with at least one repeated symbol.
It follows that
№60 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
An alternative solution to Example 3(b) is based on the observation that if S is the set of all PINs and A is the set of all PINs with no repeated symbol, then S – A is the set of all PINs with at least one repeated symbol.
It follows that
№61 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
An alternative solution to Example 3(b) is based on the observation that if S is the set of all PINs and A is the set of all PINs with no repeated symbol, then S – A is the set of all PINs with at least one repeated symbol.
It follows that
№64 слайд
Содержание слайда: Probability
The Difference Rule---Cont.
This solution illustrates a more general property of probabilities: that the probability of the complement of an event is obtained by subtracting the probability of the event from the number 1.
Formula for the Probability of the Complement of an event!
If S is a finite sample space and A is an event in S, then
P(Ac) = 1- P(A).
№65 слайд
Содержание слайда: Probability
The Inclusion/Exclusion Rule
The addition rule says how many elements are in a union of sets if the sets are mutually disjoint. Now consider the question of how to determine the number of elements in a union of sets when some of the sets overlap.
For simplicity, begin by looking at a union of two sets A and B, as shown below.
№70 слайд
Содержание слайда: Further Counting
Counting Subsets of a Set: Combinations:
Look at these examples:
In how many ways, can I select 5 books from my collection of 100 to take on vacation?
How many different ways 13-card Bridge hands can be dealt from a 52-card deck?
In how many ways, can I select 5 toppings for my pizza if there are 14 available?
What is common in all these questions?
№71 слайд
Содержание слайда: Further Counting
Counting Subsets of a Set: Combinations:
Look at these examples:
In how many ways, can I select 5 books from my collection of 100 to take on vacation?
How many different ways 13-card Bridge hands can be dealt from a 52-card deck?
In how many ways, can I select 5 toppings for my pizza if there are 14 available?
What is common in all these questions?
Each is trying to find “how many k-element subsets of an n-element set are there?”
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