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Probability definition|Addition Theorems on Probability




what is probability

Probability is the measure of the likeliness that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty)

Why we need Probability and what is the use of it?

It is widely used in the study of Mathematics, Statistics, Gambling, Physical sciences, Biological sciences, Weather forecasting, Finance etc. to draw conclusions. Insurance companies uses this to decide on financial policies

Terms related to Probability

Randomness or Random experiments

  • When we perform experiments in science and engineering repeatedly under very nearly identical conditions, we almost get the same result.
  • Such experiments are called deterministic experiments.
  • There also exists experiments When next outcome of the experiment cannot be determined  even they are performed under nearly identical conditions then we say it is a random experiment
  • e.g. Consider the dice. When we throw the dice, we cannot determine what number will come Since we cannot predict the next outcome, we may say it is a random experiment
  • Probability theory is concerned with such random phenomena or random experiments

Trial

A trial is an action which results in one or several outcomes, for example each toss of the coin and each throw of the die are called trials

Independent Trial

Successive trials of some random event for example tosses of a coin,throws of a die are said to be independent if the outcome of any one trial does not impact the outcomes of any others.

Sample space

  • It is a set of all possible outcomes of an random experiment which we defined earliar
    e.g. when we coin is tossed, the possible outcome are Head and Tail.So sample space is Head and tail
  • It is generally denoted by letter S
  • The sample space can be either be finite or infinite.
  • Example for infinite Sample space would be an experiment where we are  tossing the coin    until Head comes up.
    Sample space { (H), (TH),(TTH), (TTTH)..}
  • We can use Tree diagram to find out the sample space. Permutation and combination studied in earliar chapter also  will be used extensively to find the sample space

Elementary event

Each outcome of the random experiment is known as elementary events

Event

  • An event is a possible outcome of the Experiment. Any subset E of the sample space S is called an event.
  • Example of events
    1. Roll a die: the outcome is even {2, 4, 6}.
    2. Flip a coin twice and the two results are different:
    {(H, T),(T, H)}.

Different type of events


Elementary or simple event

If it contains the single elements of the sample space

Compound event

If it contains more than single elements of the sample space

Impossible event

If it can never occur.

Sure or certain event

An event which is sure to occur

Algebra of events operations

We know the event are sets and they are subset of Sample space. So normal set operations can be done over them also
Let A and B are two events of the Sample Space S

Union: (A ∪ B) an outcome is in A ∪ B if it is either

in A or in B.

Venn Diagram for Union of Sets

Intersection: (A ∩ B), AB: an outcome is in AB if

it is in both A and B.

Venn Diagram for Intersection of Sets

Difference (A -B)  , the event of occurrence of A but not B
Venn Diagram for Difference of Sets

Difference (B -A)  , the event of occurrence of B but not A


AC  : The event of non-occurrence of the event A.It is also called the complementary event of A




Some More Important Definition on Probability

Equally Likely Outcomes

The outcome of the random experiments are called equally likely outcomes. If all of them are equal preferences.

Exhaustive Outcomes

The outcome of the random experiments are called exhaustive  outcomes. If they cover all the possible outcomes of the experiments

Theoretical probability definition or Classical Probability

The theoretical probability or the classical probability of the event A in a random experiment having exhaustive, equal likely outcome is defined as
theoretical probability definition
  1. Let Number of exhaustive equally likely cases of the experiment =n
    Number of outcome favorable to A=m
    Then
    $P( A) = \frac {m}{n}$
    Also you may have observed that
    $ 0 \leq m \leq n$
    Or
    $ 0 \leq \frac {m}{n} \leq 1$
  2. The event $A^c$ representing ‘not A’, is called the complement of the event A. We also say that $A^c$ and $A$ are complementary events.
    Now
    $P(A^c) = \frac {n-m}{n} = 1 - \frac {m}{n} = 1- P(A)$
    Or
    $P(A) +P(A^c)=1$
  3. The probability of an event (U) which is impossible to occur is 0. Such an event is called an impossible event
    $P(U)=0$
  4. The probability of an event ( X) which is sure (or certain) to occur is 1. Such an event is called a sure event or a certain event
    $P(X)=1$
    /li>
  5. Probability of any event can be as
    theoretical probability definition
  6. Odds in favor of the happening of the event is defined as ratio of P(A) and P(Ac/sup>)
    Let it be    m: n
    Then $ \frac {P(A)} {P(A^c)}=\frac { m}{n}$
    Or
    $\frac {P(A)}{1- P(A)} =\frac {m}{n}$
    Or
    $P(A) = \frac {m}{(m + n)}$
  7. Odds against the happening of the event is defined as ratio of P(Ac) and P(A)
    Let it be m: n
    Then
    $P(A) =\frac {n}{ (m + n)}$

How to Solve Probability Problems

  1. Find out the number of elements in the sample space S of the given random experiment. It would be nice to write sample space elements also if possible
  2. Find out the elements which lies in the event (A) which we are looking to find probability. Count the elements in it
  3. Both the step 1 and step 2 depends on the complexity of the problem. We may have to use  Permutation and combination for arrangement questions
  4.  Use classical definition of probability to find out the answer

Addition Theorems on Probability

  1. Let E and F are two events  of an random experiments
    Then
    $P(E \cup F) = P(E) + P(F) - P(E \cap F) $
    I event E and F are mutually exclusive
    $n( E \cap F) =0$
    Then
    $P(E \cup F) = P(E) + P(F)$
  2. Addition Theorem of three events
    Let E ,F and G are three events  of an random experiments
    $P(E \cup F \cup G ) = P(E) + P(F) + P(G) - P(E \cap F) -  P(F \cap G) - P(E \cap G )  + P(E \cap F \cap G)$
    If event E , F and G are mutually exclusive
    $n( E \cap F) = 0$
    $n(E \cap G) = 0$
    $n(F \cap G) = 0$
    $n(E \cap F \cap G)= 0$
    Then
    $P(E \cup F\cup G ) = P(E) + P(F) + P(G)$
  3. Let E and F are two events  of an random experiments
    Then
    Probability of the occurrence of the Event E only $( E \cap F^c)$
    $ P(E \cap F^c) = P(E) - P(E \cap F )$
    Probability of the occurrence of the Event F only $( E^c \cap F)$
    $P(E^c \cap F) = P(F) - P(E \cap F )$
    Probability of occurrence of exactly one of the two event E and F
    $P[(E^c \cap F) \cup (E \cap F^c) ] = P(E) + P(F) - 2 P(E \cap F)$

Solved examples

Example 1)
E, F and G are events associated with a random experiment such that
$P(E) = 0.3, P(F) = 0.4, P(G) = 0.8$ $P(E \cap F) = 0.08 ,P( E \cap G) = 0.28 \; and \; P(E \cap F \cap G) = 0.09$. If $P(E \cup F \cup G) \geq 0.75$ then prove that $P(F \cap G)$ lies in the interval [0.23, 0.48]
Answer
We know that
$P(E \cup F \cup G ) = P(E) + P(F) + P(G) - P(E \cap F) $
$- P(F \cap G) - P(E \cap G ) + P(E \cap F \cap G)$
$P(E \cup F \cup G ) = .3 +.4+.8 -.08-.28 +.09 - P(F \cap G)$
$P(E \cup F \cup G ) =1.23 - P(F \cap G)$ -(1) Or
$1.23 - P(F \cap G) \geq 0.75$
$1.23 -.75 \geq P(F \cap G)$
$.48 \geq P(F \cap G) $
Now we know any Probability is less than 1
So
$P(E \cup F \cup G ) \leq 1$
So
$1.23 - P(F \cap G) \leq 1$
Then
$P(F \cap G) \geq .23$
So
$.23 \leq P(F \cap G) \leq .48$

Example 2)
From a pack of 52 cards, two cards are drawn together, what is the probability that both the cards are kings
Answer
Total outcome = 52C2  = 1326
Total King cases =4C2 =6
Probability ==6/1326=1/221

Example 3)
Three unbiased coins are tossed. What is the probability of getting at most two heads?
Answer
S={TTT,TTH,THT,HTT,THH,HTH,HHT,HHH}
Let E be the event of getting at most two heads
E={TTT,TTH,THT,HTT,THH,HTH,HHT}
P(E) =7/8

Example 4)
Find the probability of selecting a red card or a 7 from a deck of 52 cards?
Answer
We need to find out P(R or 7)
Probability of selecting a RED card=P(R) = $\frac {26}{52}$
Probability of selecting a 7 =P(7) = $\frac {4}{52}$
Probability of selecting both a red card and a 7
$P( R \cap 7) = \frac {2}{52}$
Now $P(R \cup 7) = P(R) + P(7) - P(R \cap 7)$
$= \frac {26}{52} + \frac {4}{52} - \frac {2}{52}$
$= \frac {28}{52}$

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