Probability.
Definition:
Probability is a measure of
chance/certainity/likelihood/possibility of occurrence of
something.
Terms associated with probability
1. Random/probability experiment/trial: A process that leads to
well defined results. Eg
• Tossing a coin
• Rolling a die
• Playing a football game
• Writing examinations
• Giving birth
Terms associated with probability
(cont…)
2. Out come: A result of a random experiment
Eg
• a win as a result of playing a game
• Pass as a result of writing an examination
• Head or tail as a result of tossing a coin.
3. Event: An outcome of interest of a random experiment
Eg. A boy as an outcome of interest of a random experiment
, giving birth.
Terms associated with probability (cont…)
4. Sample/possibility
space : A set of all possible outcomes of a
random experiment.
Sample space for a single random experiment:
Find the sample space in each of the following random
experiments:
1. Tossing a coin
2. Rolling a die
3. Writing an examination
4. Find the sample space for drawing one card from an
ordinary deck of cards.
Sample space for a single experiment
Solution
1. { Head, Tail} or {H,T}
2. {1,2,3,4,5,6}
3. {pass, fail}
Sample space for a single experiment(
cont…)
4, Since there are four suits(hearts, clubs, diamonds, and spades) and 13 cards fro each
suit ( ace through king) there are 52 outcomes in the sample space as follows:
Sample space for a paired experiments
Find the sample space in each of the following random
experiments:
1. Tossing a coin and rolling a die at the same time.
2. Rolling two dice simultaneously
3. Gender of children if a family has three children.
Use M for male and F for female.
Sample space for a paired
experiments( cont …)
Solution
1.
Rolling a die
Tossing
a
coin
1
2
3
4
5
6
H
H,1
H,2
H,3
H,4
H,5
H,6
T
T,1
T,2
T,3
T,4
T,5,
T,6
Sample space for a paired
experiments( cont …)
2.
Rolling die 1
Rolling
die 2
1
2
3
4
5
6
1
1,1
1,2
1,3
1,4
1,5
1,6
2
2,1
2,2
2,3
2,4
2,5
2,6
3
3,1
3,2
3,3
3,4
3,5
3,6
4
4,1
4,2
4,3
4,4
4,5
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
6
6,1
6,2
6,3
6,4
6,5
6,6
Sample space for a paired experiments( cont
…)
3.
• After the first birth {M,F}
• After the second birth
First birth
Second
birth
M
F
M
M,M
M,F
F
F,M
F,F
• After the third birth
First two births
Third
birth
M,M
M,F
F,M
F,F
M
M,M,M
M,M,F
M,F,M
M,F,F
F
F,M,M
F,M,F
F,F,M
F,F,F
Finding sample space using a tree diagram
Tree diagram: Is a diagram consisting of line segments
emanating from a starting point and also from the outcome
point.
Eg
2nd Exp
outcome
1st Exp
outcome
2nd Exp
Note: a branch represents outcome and a point of branching represents experiment.
Finding sample space using tree
diagram
Sample space of gender of children if a family has three children. Use M for
male and F for female
Types of probability
The types of probability include the following
1. Subjective probability: Thus a probability derived from one’s personal
judgment/experience about whether an event is likely going to occur.
Eg
•
There is 90% chance that team A will win the game because last year and
last of last year it worn against the same team.
•
If you propose her there is 5% chance that she will reject you.
Disadvantage: Not accurate
Types of probability
2. Objective(Empirical) probability : Probability derived from analysis/calculation
based on recorded observation of a random experiment
Eg
•
Probability of getting a head after tossing a coin.
•
Probability of getting a 4 after rolling a die.
•
Probability of getting an ace after drawing a playing card.
3. Classical probability : Thus a special case of objective probability which involves
experiments whose outcomes are equally likely ( have the same chance of
occurring)
Examples of such experiments include
•
Tossing a coin, rolling a die, giving birth.
Calculating probability of an event
Probability that an event E will occur is given by
P(E) =
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑬
𝑠𝑖𝑧𝑒 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑎𝑝𝑐𝑒(𝑺)
=
𝑛(𝑬)
𝑛(𝑺)
Note: Probability of an event E can be presented as a
• Fraction or
• Decimal or
• Percentage
Calculating probability of an event
Example:
Find probability of;
a) getting an even number from S= { 1,2,3,4,5,6,7,8,9}
b) getting a jack after drawing a card from a deck of standard playing cards
c) getting the sum of the outcome being less than 8 if two dice are rolled at
the same time.
d) Travelling by driving given the following methods of travelling
Method
Frequency( how many used it)
By driving
41
By flying
6
Others means
3
Total
50
Calculating probability of an event
Solution
1. P(even number) =
2. P( jack card)
𝑛(𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟)
𝑛(𝑆)
𝑛(𝑗𝑎𝑐𝑘 𝑐𝑎𝑟𝑑𝑠)
=
𝑛(𝑆)
=
=
4
9
4
52
3. Sample space
Die 1
Die 2
1
2
3
4
5
6
1
1+1=2
1+2=3
1+3=4
1+4=5
1+5=6
1+6=7
2
2+1=3
2+2=4
2+3=5
2+4=6
2+5=7
2+6=8
3
3+1=4
3+2=5
3+3=6
3+4=7
3+5=8
3+6=9
4
4+1=5
4+2=6
4+3=7
4+4=8
4+5=9
4+6=10
5
5+1=6
5+2=7
5+3=8
5+4=9
5+5=10
5+6=11
6
6+1=7
6+2=8
6+3=9
6+4=10
6+5=11
6+6=12
Calculating probability of an event
Solution
3. P( sum < 8) =
𝑛(𝑠𝑢𝑚 < 8)
𝑛(𝑆𝑢𝑚)
4. P( by driving) =
21
= 36
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑏𝑦 𝑑𝑟𝑖𝑣𝑖𝑛𝑔
𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
41
= 50
Basic Probability Rules
 The probability of any event,𝐸,is a number (either a fraction
or a decimal) between and including 0 and1 .i,e 0≤P(𝐸)≤1
 If an event,𝐸,can not occur (i.e.the event contains no
members in the sample space) its probability is 0.
 If an event ,𝐸, is certain, then the probability is 1.
 The sum of all probabilities of all out comes in the sample
space is 1. i.e P(S)= 1
 Probability of a complement of an event is given by
P(Ec) or P(E’) or P(𝑬) = 1 – P(E)
Mutually exclusive events
Events are said to be mutually exclusive if the occurrence of one event means
that the other event can not occur. In this case when one event takes place,
the probability of the other event occurring is zero.
Example
When a die is rolled, which of the following events are mutually exclusive
Event A: getting a 1
Event B : getting a 4
Event C: getting an even number
Event B: getting a numbe less than 5
Independent events
Events are said to be independent if the occurrence of one event doe not
affect the occurrence of another.
Example
Give examples of independent events of a given random experiment
Additional rule of probability
Given events A and B. The probability of having event A or event B occurring is
given by
P( A or B) = P(A) + P(B) – P( A∩B)
This equation is called the additional rule of probability
Note: If A and B and Mutually exclusive, P( A∩B) = 0.
Example
When a die is rolled find the probability of getting:
a) An even number or number less than 5
b) An even number or odd number
Additional rule of probability
Solution
S = { 1, 2, 3, 4, 5, 6}
Let even number be event A, number less than 5 be event B, odd number be
event C
a) P(A or B) = P(A) + P(B) – P( A∩B) = 3/6 + 4/6 - 2/6 = 5/6
b) P(A or c) = P(A) + P(C) – P( A∩ C) = 3/6 + 3/6 - 0 = 6/6 =1
Multiplication rule of probability
Given event A and event B, the probability of occurrence of both the events
A and B is given by:
P(A ∩ B) = P(A) . P(B) if A and B are two independent events
P(A and B) = P(A ∩ B) = P(B) . P(A|B) If A and B are dependent events,
otherwise its given by
Example
A box contains 20 red and 10 blue balls. Two balls are drawn from a bag one
after the other with replacement. What is the probability that the balls drawn
are red and blue.
i)
In that order
ii) In any order
Multiplication rule of probability
Solution: Let red ball be even A and blue ball be even B P(A∩B) or P(AB)
if A and B are two independent events
i)
In that order
P( A and B) = P(A ∩ B) = P(A) . P(B) = 20/30 . 10/30 = 2/9
i)
In any order
P( A and B or B and A) = P(A ∩ B) +(B ∩A) = 2[P(A) . P(B)] = 2[20/30 . 10/30] = 4/9
Conditional probability
In probability theory, conditional probability is a measure of the probability of
an event occurring, given that another event has already occurred. Let A and
B be some events the probability of A occurring given that B has already
occurred is given by
P(A|B) =
𝑃(𝐴 ∩𝐵)
𝑃(𝐵)
Example
Two dies are thrown simultaneously. What is the probability that the number 3
has appeared at least once given that the sum of the numbers obtained is
found to be 7.
Conditional probability
Soln
Sample space is ?
Let event A indicates the combination in which 3 has appeared at least once.
And event B indicates the combination of the numbers which sum up to 7.
A = {(3, 1), (3, 2), (3, 3)(3, 4)(3, 5)(3, 6)(1, 3)(2, 3)(4, 3)(5, 3)(6, 3)}
B = {(1, 6),(2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
A∩B=2
P(B) = 6/36
P(A ∩ B) = 2/36
Therefore P(A|B) = P(A∩B)/P(B) = (2/36)/(6/36) = ⅓