Mathematics ADDITION AND MULTIPLICATION THEOREMS OF PROBABILITIES

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ADDITION AND MULTIPLICATION THEOREMS OF PROBABILITIES

Addition Theorem On Probability :

If `A` and `B` are two events associated with an experiment then `P(A cup B)` is called the sum of the probabilities of all the sample points in `A cup B` or probability of occurrence of atleast one of the events from `A` and `B` and the expression for `P(A cup B)` is called the addition theorem on probability From the Venn diagram it is clear that

`P`(Occurence at least one of the events from `A` and `B`) `P`(`A` or `B` or both) or `P(A + B)`
`=>P(A cup B)=P(A)+P(B)-P(A cap B)`
`=P(A)+P(bar A cap bar B)`
`=P(B)+(A cap bar B)`
`=P(A cap bar B)+P(AB)+P(bar A cap B)`
`=1-P(bar A cap B)`
`=1-P(bar(A cap B))`

`text(Note :)`

(i) If A and Bare mutually exclusive events then
`p(AuuB) = p(A) + p(B)` `\ \ \ \ {because p(AnnB) = 0`}
(ii) If A and Bare exhaustive events then `P(A uu B) = 1`
(iii) `P(A uu B) = 1 - bar(P(A uu B))`

If `A` and `B` are two events associated with an experiment then `P(A cup B)` is called the sum of the probabilities of all the sample points in `A cup B` or probability of occurrence of atleast one of the events from `A` and `B` and the expression for `P(A cup B)` is called the addition theorem on probability From the Venn diagram it is clear that

`P`(Occurence at least one of the events from `A` and `B`) `P`(`A` or `B` or both) or `P(A + B)`
`=>P(A cup B)=P(A)+P(B)-P(A cap B)`
`=P(A)+P(bar A cap bar B)`
`=P(B)+(A cap bar B)`
`=P(A cap bar B)+P(AB)+P(bar A cap B)`
`=1-P(bar A cap B)`
`=1-P(bar(A cap B))`

`text(Note :)`

(i) If A and Bare mutually exclusive events then
`p(AuuB) = p(A) + p(B)` `\ \ \ \ {because p(AnnB) = 0`}
(ii) If A and Bare exhaustive events then `P(A uu B) = 1`
(iii) `P(A uu B) = 1 - bar(P(A uu B))`

Multiplication Theorem On Probability :

Let `A` and `B` be two events associated with a sample space `S`. Clearly, the set `A cap B` denotes the event that both `A` and `B` have occurred. In other words, `A cap B` denotes the simultaneous occurrence of the events `E` and `F.` The event `A cap B` is also written as `AB`.

We know that the conditional probability of event `A` given that B has occurred is denoted by `P(A | B)` and is given by

`P(A|B)=(P(A cap B))/(P(B)),P(B) ne 0`

From this result, we can write

`P(A cap B)=P(B) .P(A|B)`...............(i)

Also, we know that

`P(B|A)=(P(A cap B))/(P(A)),P(A) ne 0`

`P(B|A)=(P(A cap B))/(P(A))` (since `A cap B=B cap)`

Thus, `P(A cap B)=P(A) . P(B|A)`.................(ii)

Combining `(i)` and `(ii)`, we find that

`P(A cap B) = P(A) P(B | A) = P(B) P(A | B)` provided `P(A) ne 0` and `P(B) ne 0`

The above result is known as the multiplication rule of probability.

`text(Note :)`

If A& B are independent events then `p(A/B) = P(A)` and `p(B/A)= P(B)` and in this case multiplication theorem `P(A nn B) = P(A) · P(B).`

Let `A` and `B` be two events associated with a sample space `S`. Clearly, the set `A cap B` denotes the event that both `A` and `B` have occurred. In other words, `A cap B` denotes the simultaneous occurrence of the events `E` and `F.` The event `A cap B` is also written as `AB`.

We know that the conditional probability of event `A` given that B has occurred is denoted by `P(A | B)` and is given by

`P(A|B)=(P(A cap B))/(P(B)),P(B) ne 0`

From this result, we can write

`P(A cap B)=P(B) .P(A|B)`...............(i)

Also, we know that

`P(B|A)=(P(A cap B))/(P(A)),P(A) ne 0`

`P(B|A)=(P(A cap B))/(P(A))` (since `A cap B=B cap)`

Thus, `P(A cap B)=P(A) . P(B|A)`.................(ii)

Combining `(i)` and `(ii)`, we find that

`P(A cap B) = P(A) P(B | A) = P(B) P(A | B)` provided `P(A) ne 0` and `P(B) ne 0`

The above result is known as the multiplication rule of probability.

`text(Note :)`

If A& B are independent events then `p(A/B) = P(A)` and `p(B/A)= P(B)` and in this case multiplication theorem `P(A nn B) = P(A) · P(B).`

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