Partition of a sample space and Theorem of total probability (Probability 7 NCERT)

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Partition of a sample space

A set of events E1, E2, …, En is said to represent a partition of the sample space S if 

p1 1

In other words, the events E1, E2, …, En represent a partition of the sample space S if they are pairwise disjoint, exhaustive and have nonzero probabilities. 
As an example, we see that any nonempty event E and its complement E′ form a partition of the sample space S since they satisfy E ∩ E′ = φ and E ∪ E′ = S. 

p2 1

From the Venn diagram in Fig 13.3, one can easily observe that if E and F are any two events associated with a sample space S, then the set {E ∩ F′, E ∩ F, E′ ∩ F, E′ ∩ F′} is a partition of the sample space S. It may be mentioned that the partition of a sample space is not unique. There can be several partitions of the same sample space.

Theorem of total probability

We shall now prove a theorem known as Theorem of total probability.

Theorem of total probability 
Let {E1, E2,…,En} be a partition of the sample space S, and suppose that each of the events E1, E2,…, Enhas nonzero probability of occurrence. Let A be any event associated with S, then 

p3 1

Proof Given that E1, E2,…, En is a partition of the sample space S (Fig 13.4). Therefore, 

p4 1

 Now, by multiplication rule of probability, we have 

p6 1 Therefore,

p7 1

 or        

p8 2

Example A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time. 
Solution Let A be the event that the construction job will be completed on time, and B be the event that there will be a strike. We have to find P(A). 
We have 

p9 3

Since events B and B′ form a partition of the sample space S, therefore, by theorem on total probability, we have 

p10 2

Thus, the probability that the construction job will be completed in time is 0.488.