Laws of probability and conditional probability

 Laws of probability and conditional probability

As in last post we have seen sample space and events in probability,today we will see some rules of probability and conditional probability 

To see laws of probability we will require some knowledge about independence and some terminologies regarding events.

1)Exhaustive events:The total number of possible outcomes of random experiment is known as the exhaustive events.

Example:If we roll a die,there are 6 exhaustive cases since any number can appear uppermost

2)Favorable events:The outcome which make necessary the happening of an event in a trial are called favorable events.

Example:let us consider two dies are rolled,then the number of favorable events getting sum 3 is 2,that is (1,2),(2,1)

3)Mutually exclusive events(Disjoint events):The events are said to be mutually exclusive or disjoint if they cannot both occur at same time.

Example:Outcome of single coin toss (either head or tail)

4)Equally likely events:Equally likely events are events that have same theoretical probability of occurring.

Examples:If we roll a die,all six faces have same probability of occurrence.

5)Independent events:The event in which occurrence of one does not affect the probability of occurrence of the other are called independent events.

Example:consider die is thrown twice,the result of first throw does not affect result of second throw.

Laws or rules of probability

Additive law of probability

Case 1: When events are mutually exclusive 

When two events A and B are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event.

P(A or B) = P(A) + P(B)

Problem: A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?

Probabilities: 

P(2)	=	1
		6
P(5)	=	1
		6
P(2 or 5)	=	P(2)	+	P(5)
	=	1	+	1
		6		6
	=	2
		6
	=	1
		3

Case 2: When events are non-mutually exclusive 

A and B are non-mutually exclusive the probability that A or B will occur is:

P(A or B) = P(A) + P(B) - P(A and B)

Problem: In a class of 20 students, 11 are boys and 9 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?

Probabilities: P(girl or A) = P(girl) + P(A) - P(girl and A)

	=	9	+	9	-	5
		20		20		20
	=	13
		20

Rule of Subtraction 

The probability that event A will occur is equal to 1 minus the probability that event A will not occur.

P(A) = 1 - P(A')

Suppose, for example, the probability that today will be rain is 0.80. What is the probability that today will be no rain Based on the rule of subtraction, the probability that today will be no rain is  1.00 - 0.80 or 0.20.

Rule of Multiplication 

The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.

P(A ∩ B) = P(A) P(B|A)

Problems:An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?

 Let A = the event that the first marble is black and 

let B = the event that the second marble is black. We know the following:In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, 

P(A) = 4/10.After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, 

P(B|A) = 3/9.

Therefore, based on the rule of multiplication:

P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15 = 0.133

 Conditional probability

we have seen that event A is subset of sample space that is for event A ,we have reference set is sample space.but the reference set may be another event B which is also subset of sample space.

The probability of event A given that event B is already occurred ,this is called conditional probability.It is denoted by P(A/B).

In other words,the probability of second event depends on probability of first event is called conditional probability.

Let A and B be two events defined on sample space of a random experiment,then

The conditional probability of A given B is denoted by P(A/B)and is defined as

P(A/B)=P(A∩ B)/P(B) 

Where P(B)>0

And

The conditional probability of B given A is P(B/A) and is defined as

P(B/A)=P(A∩B)/P(A)

Where P(A)>0 And

 P(A∩B)=P(B∩A)

Note: if A and B are independent event then P(A∩B)=P(A).P(B) which implies P(A/B)=P(A) and 

P(B/A)=P(B).

Example:

If we rolls two six-sided dice,and we wish to compute the probability that the value on first die is 4,given the information that their product is no greater than 8.

Let D1 and D2 be value rolled on die 1 and die 2 respectively.

Here sample space contain 36 elements . therefore each element having probability 1/36.

There 6 elements ,which has 4 number on first die.

P(D1=4)=6/36.

And there are 16 elements present which has D1.D2<=8.

P(D1.D2<=8)=16/36.

There are 2 elements which has 4 on first die and D1.D2<=8.

P(D1=4/D1.D2<=8)=2/16=1/8.

           Or

P(D1=4/D1.D2<=8)=P(D1=4#D1.D1<=8)/P(D1.D2<=8)

  =(2/36)/(16/36)=1/8.

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