Theorem Of Total Probability
The theorem of total probability is useful to find the probability of happening an event from the different partitions of the sample space. For a sample space divided into n partitions {E_{1}, E_{2, }E_{3}, ......E_{n}} such that {E_{1} U E2 U E_{3}, .....U E_{n}} = S, the probability of happening of the event A from the different partitions is P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n}).
The theorem of total probability is the foundation of Baye's theorem. Let us learn more about the proof and application of the theorem of total probability.
What Is the Theorem Of Total Probability?
The theorem of total probability is useful to compute the probability of happening of an event, which is the result of the probabilities of happening of this event from the different partitions of the sample space. Here we assume the sample space S to be divided into different partitions {E_{1}, E_{2, }E_{3}, ......E_{n}} such that {E_{1} U E2 U E_{3}, .....U E_{n}} = S, and the probability of happening of an event A is the summation of the probability of happening of this event from the different partitions of the sample space.
This can be better understood with the help of the below three simple examples.
 The doctor reaches the patient for treatment, takes different modes of transportation, and his probability of arriving on time is equal to the summation of his probability of arriving on time, from different modes of transportation.
 A student is to represent the school in an external competition. The probability of selecting the student for the competition is equal to the summation of the probability of selecting this student from the different classes in the school.
 The probability of finding a defective mango is the summation of the probability of finding this defective mango from the different boxes of mangoes.
The theorem of total probability is the foundation of Baye's Theorem, and it helps in deriving the reverse probability of happening of an event from partitions of a sample space S. The happening of an event from the partition of the defined space can be calculated using the theorem of total probability, and the reverse probability from the particular partition of the sample space, for the given probability of the event can be calculated with the help of Baye's Theorem, and this is completed derived using the theorem of total probability.
Statement Of Theorem Of Total Probability: The events E_{1}, E_{2, }E_{3}, ......E_{n} is a set of exhaustive events of a sample space S, such that {E_{1}, E_{2, }E_{3}, ......E_{n}} be the partitions of a sample space S, and the happening of the event A from the sample space S is P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n})
Proof Of Theorem Of Total Probability
Given that E_{1}, E_{2, }E_{3}, ......E_{n} is a set of the exhaustive event of the sample space S.
{E_{1} U E2 U E_{3}, .....U E_{n}} = S
Also_{ }E_{1}, E_{2, }E_{3}, ......E_{n }mutually exclusive events. These are events such that the happening of one event prevents the happening of another set of events, and there is no common events.
E_{i} n E_{j} = ∅
Let us consider an event A which is part of the sample space S.
A = A n S
A = A n {E_{1} U E2 U E_{3}, .....U En}
A = {A n E_{1}} U {A n E2} U {A n E_{3}}, .....U {A n En}
Let us apply probability on both sides of the above equation.
P(A) = P(A n E_{1}) U P(A n E_{2}) U P(A n E_{3}), .....U P(A n E_{n})
P(A) = P(A n E_{1}) + P(A n E_{2}) + P(A n E_{3}), .....U P(A n E_{n})
P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n})
Application Of Theorem Of Total Probability
This theorem of total probability is the foundation of Baye's Theorem. The Baye's Theorem is defined for a sample space S containing a set of events E_{1}, E_{2, }E_{3}, ......E_{n}, which together constitutes the sample space S, such that E_{1} U E2 U E_{3}, .....U E_{n} = S. The events are pairwise disjoint, exhaustive, and with nonzero probabilities. Here for event A, we have the following expression of Baye's Theorem.
P(E_{i}/A) = \(\dfrac{P(E_{i})P(A/E_{i})}{P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n})}\)
Here in the above formula, we have used the multiplication rule of probability in the numerator and the concept of a theorem of total probability in the denominator.
The multiplication rule of the probability of happening of an event from a partition of a sample space is as follows.
P(A n E_{i}) = P(A).P(E_{i}/A)
P(E_{i}/A) = \(\dfrac{P(An E_i)}{P(A)}\)
P(E_{i}/A) = \(\dfrac{P(E_{i})P(A/E_{i})}{P(A)}\)
Here we apply the abovederived formula from the theorem of total probability of P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n}) to obtain the final expression of baye's theorem of reverse probability.
P(E_{i}/A) = \(\dfrac{P(E_{i})P(A/E_{i})}{P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n})}\)
Related Topics
The following topics help in a better understanding of the theorem of total probability.
Examples on Theorem Of Total Probability

Example 1: The executive of a company uses either a train or a bus to reach his office. The probability of his being late if he travels by train is 0.1, and the probability of his being late if he travels by bus is 0.2. What is the probability that the executive reaches late to office? Use the concepts from the theorem of total probability to find the required solution.
Solution:
The probability of being late by train is P(Train_{L}) = 0.1
The probability of being late by bus is P(Bus_{L}) = 0.2
The probability of travelling by train is P(Train) = 0.5
The probability of travelling by bus is P(Bus) = 0.5
The probability of reaching office late by the executive P(Late) = P(Train).P(Train_{L}) + P(Bus).P(Bus_{L})
= 0.5 x 0.1 + 0.5 x 0.2
= 0.05 + 0.10
= 0.15
Therefore the probability of reaching late to the office by the executive is 0.15.

Example 2: Using the concept from the theorem of total probability, find the probability of finding a raw mango from two baskets, if the probability of picking a raw mango from one basket is 1/8 and the probability of finding a raw mango from the second basket is 1/10.
Solution:
The probability of picking a raw mango from the first basket is P(B_{1}) = 1/8
The probability of picking a raw mango from the second basket is P(B_{2}) = 1/10
The probability of taking the first basket is P(I) = 1/2
The probability of taking the second basket is P(II) = 1/2
Let us now use the concept from the theorem of total probability to find the probability of picking a raw mango from the two baskets.
P(Raw Mango) = P(I).P(B_{1}) + P(II).P(B_{2})
= 1/2.1/8 + 1/2.1/10
= 1/16 + 1/20
= (5 × 1)/(5 × 16) + (4 × 1)/(4 × 20)
= 5/80 + 4/80
= (5 + 4)/80
= 9/80
Therefore, the probability of finding the raw mango is 9/80.
FAQs on Theorem Of Total Probability
What Is The Theorem Of Total Probability?
The theorem of total probability is useful to find the probability of happening an event from the different partitions of the sample space. For a sample space divided into n partitions {E_{1}, E_{2, }E_{3}, ......E_{n}} such that {E_{1} U E2 U E_{3}, .....U E_{n}} = S, the probability of happening of the event A from the different partitions is P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n}).
What Is The Formula Of Theorem Of Total Probability?
The formula of the probability of happening of event A from the different partitions is P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n}). This formula is useful to find the total probability of the event from the different partitions of the sample space.
What Is The Difference Between Theorem Of Total Probability And Baye's Theorem?
The theorem of total probability defines the probability of happening of an event from the different partitions of the sample space and the baye's theorem defines the reverse probability of happening of the of the event from a particular partition of the sample space. The formula for the theorem of total probability is P(A) = P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n}), and the formula for the Baye's theorem is
P(E_{i}/A) = \(\dfrac{P(E_{i})P(A/E_{i})}{P(E_{1})P(A/E_{1}) + P(E_{2})P(A/E_{2}) + ......P(E_{n})P(A/E_{n})}\).
What Is The Application Of Theorem Of Total Probability?
The theorem of total probability is prominently applied to derive Baye's theorem, which is the reverse probability of happening of find the partition of the sample space from which the event has happened.
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