Transcript Definition: Properties of frequency

§2 Frequency and probability
2.1The definitions and properties of frequency and properties
1.The definition and properties of frequency
Definition:
Consider performing our experiment a large number ,n times and counting
the number of those times when A occurs. The frequency of A is then
defined to be
f n ( A) 
nA Number of times when A occurs

n
Number of total trials
Properties of frequency:
Example: A coin is tossed 5times、50times、500time，this
experiment is repeated 7 times .Observe the number of Head appear
and frequency.
f (H )
1
.
2
With the increase of n, the frequency f presents stability
Probability
Probability of an event A of a repeatable experiments is given by
nA
Number of times when A occurs
P( A)  lim
 lim
N  n
N 
Number of total trials
Experiment ‘tossing a coin’:The relative frequency of the event ‘head up’
as the function of the number of trials.
Probability Axioms
Definition 1.9
A Probability measure on a sample space S is a function P which assigns a number
P(A) to every event A in S in such a way that the following three axioms are
satisfied:
Axiom 1. P(A) ≥0 for every event A.
Axiom 2. P(S)=1.
Axiom 3. Countable additivity可列可加性. i.e. , if A1,A2,…… is an infinite sequence
无穷序列 of mutually exclusive (disjoint) event两两互不相容事件 then


P ( Ai )   P( Ai )
i 1
i 1
or P ( A1  A2  )  P ( A1 )  P ( A2 ) 
Properties of Probability
1. P(Ø) = 0,P(S) = 1. 0  P( A)  1
2.If A and B are disjoint events then
(disjoint or mutually exclusive means A∩B = Ø)
3.For any event A, P( A ) = 1 – P(A).
4.If B  A then P(A - B) = P(A) - P(B) and P( B)  P( A)
5.For any A and B,
P(A∪B ) = P(A) + P(B) − P(A∩B )
P(A∪B )  P(A) + P(B)
B
A
2.2 Equally Likely Outcomes
The outcomes of a sample space are called equally likely if all of them
have the same chance of occurrence. It is very difficult to decide whether or
not the outcomes are equally likely. But in this tutorial we shall assume in
most of the experiments that the outcomes are equally likely. We shall apply
the assumption of equally likely in the following cases：
(1) Throw of a coin or coins:
When a coin is tossed, it has two possible outcomes called
head and tail. We shall always assume that head and tail are
equally likely if not otherwise mentioned. For more than one coin,
it will be assumed that on all the coins, head and tail are equally likely
(2) Throw of a die or dice:
Throw of a single die can be produced six possible outcomes.
All the six outcomes are assumed equally likely. For any number
of dice, the six faces are assumed equally likely.
(3) Playing Cards:
There are 52 cards in a deck of ordinary playing cards.
All the cards are of the same size and are therefore assumed
equally likely.
If an experiment has n simple outcomes, this method
would assign a probability of 1/n to each outcome. In
other words, each outcome is assumed to have an equal
probability of occurrence.
This method is also called the axiomatic approach.
Example 1: Roll of a Die
S = {1, 2, · · · , 6}
Probabilities: Each simple event has a 1/6 chance of occurring.
Example 2: Two Rolls of a Die
S = {(1, 1), (1, 2), · · · , (6, 6)}
Assumption: The two rolls are “independent.”
Probabilities: Each simple event has a (1/6) · (1/6) =1/36 chance
of occurring.
nA
Number of times when A occurs
P( A)  lim
 lim
N  n
N 
Number of total trials
Theorem1.1
In classical probability counting is used for calculating
probabilities. For the probability of an event A we need
to know the number of outcomes in A, k, and
if the sample space consists of a finite number of
equally likely outcomes, also the total number of
outcomes, n.
P( A) 
k Number of elements in A

n Number of elements in S
P( A) 
E1:
k Number of elements in A

n Number of elements in S
A spinner has 4 equal sectors colored yellow, blue, green and red.
After spinning the spinner, what is the probability of landing on each
color?
P(yellow) =
P(blue) =
Number of ways to land on yellow
Total number of colors
Number of ways to land on blue
Total number of colors
Number of ways to land on green
P(green) =
Total number of colors
P(red) =
Number of ways to land on red
1

Total numberof colors
4


1
4
1
4
1

4
E2:
A single 6-sided die is rolled. What is the probability of each outcome?
What is the probability
of rolling an even number? of rolling an odd number?
Roll of a Die
P(even) = 3/6
P(low) = 3/6
P(even and low) = P({2}) = 1/6
P(even or low) = 3/6 + 3/6 − 1/6 = 5/6
P({1} or {6}) = 1/6 + 1/6 − 0 = 2/6
E3:
A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles.
If a single marble is chosen at random from the jar, what is the
probability of choosing a red marble? a green marble? a blue marble?
a yellow marble?
P(red) =
number of ways to choose red
6
3


Total number of marbles
22 11
P(green) =
P(blue) =
number of ways to choose green
5

Total number of marbles
22
number of ways to choose blue
Total number of marbles
P(yellow) =
number of ways to choose yellow
Total number of marbles

8
4

22 11

3
22