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Total Probability Theorem is a fundamental theorem in Probability that relates the probability of a composite event to the probabilities of its sub-events. The theorem states that if C1, C2, ..., Cn is a partition of the sample space S, then the probability of any event A is equal to the sum of the probabilities of the sub-events of A.
Total Probability Theorem is stated as P(A)= Σnk=0 P(Ck) P(A|Ck)
Total Probability Theorem is the foundation of Bayes Theorem.
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Key Terms: Total Probability Theorem, Probability, Law of Probability, Event, Sample Space, Bayes Theorem, Conditional Probability
What is Total Probability Theorem?
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Total Probability Theorem states that for a sample space that is composed of a finite or countable number of mutually exclusive events, the probability of the sample space is equal to the sum of the probabilities of the individual events.
- In mathematical terms, if S is the sample space and A1, A2, A3, ..., An are mutually exclusive events that make up S, then the probability of S is given by the equation P(S) = P(A1) + P(A2) + P(A3) + ... + P(An).
- This theorem is also known as the Law of Total Probability.
- It is used in conditional probability and Bayes' theorem and also to find the marginal probability of an event from the joint probability distribution.
- Total Probability Theorem is used in many areas, including statistics, signal processing, and decision theory.
- This theorem helps in finding the conditional probability of an event when the sample space is divided into a number of mutually exclusive events.
- One of the most common applications of Total Probability Theorem is Bayes' theorem, which is used in machine learning and data analysis.
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| Chance And Probability | Types of Events in Probability | Probability Distribution Formula |
| Probability & Statistics | Probability and Statistics Symbols | Theoretical Probability |
Law of Total Probability
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Law of Total Probability states that if there are two events A and B in a sample space S, then such space can be referred to as A ∩ B′, A ∩ B, A′ ∩ B, A′ ∩ B′.. This expression reflects the total probability realized and these sets are mutually exclusive
- Law of Total Probability, also known as the "Total Probability Theorem" is a fundamental concept in probability theory.
- It states that for any event B in a sample space S, the probability of B can be calculated by summing the probabilities of the sub-events of B that belong to each mutually exclusive event Ai in the sample space.
- Law of Total Probability is often written as P(B) = P(B|A1)P(A1) + P(B|A2)P(A2) + ... + P(B|An)P(An).
Law of Total Probability Statement
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The proof of the total probability theorem is relatively straightforward. The first step is to recognize that the sample space, S, is made up of the mutually exclusive events A1, A2, A3, ..., An. This means that if an event occurs, it can only occur in one of the individual events and not in any of the others.
- The theorem is based on the principle of mutually exclusive events, which means that the events Ai can't happen at the same time.
- This law helps in finding the overall probability of an event by breaking it down into smaller, mutually exclusive sub-events.
- It is useful in many areas of statistics, signal processing, decision theory and machine learning, and is often used in conjunction with Bayes' theorem.
Total Probability Theorem
Let events C1, C2 . . . Cn form partitions of the sample space S, in which all the events have a non-zero probability of occurrence. According to the total probability theorem, for any event, A associated with S
P(A)= Σnk=0 P(Ck) P(A|Ck)
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Total Probability Theorem Proof
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{C1, C2, . . . . , Cn} is the partitions of the sample space S such that, Ci ∩ Ck = φ, where i ≠ k and i, k = 1, 2,…,n also all the events C1, C2 . . . . Cn have non zero probability. Sample space S can be given as,
S = C1 ∪ C2 ∪ . . . . . ∪ Cn
For any event A,
A = A ∩ S
= A ∩ (C1 ∪ C2 ∪ . . . . ∪ Cn)
= (A ∩ C1) ∪ (A ∩ C2) ∪ … ∪ (A ∩ Cn) . . . . . (1)
We know that A ∩ Ci and A ∩ Ck are the subsets of Ci and Ck. Here, Ci and Ck are disjoint for i ≠ k. since they are mutually independent events which implies that A ∩ Ci and A ∩ Ck are also disjoint for all i ≠ k. Thus,
P(A) = P [(A ∩ C1) ∪ (A ∩ C2) ∪ ….. ∪ (A ∩ Cn)]
= P (A ∩ C1) + P (A ∩ C2) + … + P (A ∩ Cn) . . . . . . . (2)
We know that,
P(A ∩ Ci) = P(Ci) P(A|Ci) (By multiplication rule of probability) . . . . (3)
Using (2) and (3), (1) can be rewritten as,
P(A) = P(C1)P(A| C1) + P(C2)P(A|C2) + P(C3)P(A| C3) + . . . . . + P(Cn)P(A| Cn)
Hence, the theorem can be stated in form of equation as,
P(A)= Σnk=0 P(Ck) P(A|Ck)
To understand this in a simple manner:
- Now, consider the probability of the sample space, P(S).
- By the definition of probability, P(S) represents the sum of probabilities of all possible outcomes in the sample space.
- Since the events A1, A2, A3, ..., An are mutually exclusive and cover the entire sample space, the probability of the sample space is the sum of the probabilities of the individual events:
P(S) = P(A1) + P(A2) + P(A3) + ... + P(An)
- This equation demonstrates that the total probability of the sample space is equal to the sum of the probabilities of the individual events that make up the sample space. This is the total probability theorem.
- It can also be visualized by Venn diagrams where the sample space is represented by a big circle with the events represented by smaller circles inside the big one and all these smaller circles combined makes the big circle.
- The area of each smaller circle is the probability of that event.
Total Probability Theorem Example
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Example: There is a factory X whose bulbs work over 5000 hours 99% of the time, while for factory it’s the same time with 95%. Factory X produces 60% of the bulbs while Y produces the rest 40%. We have to calculate the probability of a bulb working over 5000 hours:
Solution. According to the Total Probability Theorem, we have
P(A)= P(A|Bx).P(Bx)+P(A|By).P(By)
i.e. 99/100 . 6/10 + 95/100 . 4/10 = 974/1000
Therefore the probability is 97.4%
Check More:
| Related Topics | ||
|---|---|---|
| Empirical Probability Formula | Conditional Probability Formula | Independent Events in Probability |
| Joint Probability | Experimental Probability | Bayes Theorem Formula |
Things to Remember
- Total Probability Theorem is also known as the Law of Total Probability,.
- According to the law,if a sample space which consists of a limited number of mutually exclusive events, then the probability of such sample space is equal to the sum of individual probabilities of individual events.
- Equation of Total Probability Theorem is P(A)= Σnk=0 P(Ck) P(A|Ck).
- It is necessary that the events considered are mutually exclusive, which means they cannot overlap with each other and they must make up the whole of the sample space.
- Total Probability Theorem can also be used to find the marginal probability of a particular event from conditional probability and joint probability.
- It is the basic foundation of Bayes theorem, Markov Chains, and the Decision theory.
Previous Years’ Questions
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Sample Questions
Ques. What is the Total Probability Theorem? (5 Marks)
Ans. Total Probability Theorem, also known as the Law of Total Probability, is a fundamental concept in probability theory. It states that the overall probability of a particular event can be calculated by summing the individual probabilities of all possible outcomes that could lead to that event. Here are the key points of the Total Probability Theorem:
- Partition of Sample Space: The theorem of total probability starts by partitioning the sample space, which is the set of all possible outcomes of an experiment, into mutually exclusive and collectively exhaustive sub-events. This means that each sub-event should represent a different and distinct possibility, and together, they should include all possible outcomes.
- Probability of Each Sub-Event: The next step is to calculate the probability of each sub-event, which is the fraction of favorable outcomes over the total number of possible outcomes in that sub-event.
- Total Probability of an Event: Once the probabilities of each sub-event are calculated, the total probability of an event can be calculated by summing the probabilities of those sub-events that could lead to that event.
- Conditional Probability: The theorem of total probability can also be used to calculate the conditional probability of an event, given that another event has occurred.
- Applications: The theorem of total probability has numerous applications in fields such as statistics, machine learning, data analysis, and decision making, where it is used to calculate the probability of a given event based on information from multiple sources.
Ques. What is the Formula of the Theorem of Total Probability? (5 Marks)
Ans. The theorem of total probability is represented by the following formula:
P(A) = ∑ P(A|B) * P(B)
Where
- P(A) is the overall probability of event A.
- P(A|B) is the conditional probability of event A given that event B has occurred.
- P(B) is the probability of event B.
- The summation (∑) is taken over all possible sub-events B that partition the sample space.
The overall probability of event A is equal to the sum of the probability of each sub-event B multiplied by the conditional probability of event A given that event B has occurred.
- This formula allows us to calculate the total probability of an event by considering the individual probabilities of all possible outcomes that could lead to that event.
- The theorem of total probability is a useful tool for analyzing complex probability distributions and making informed decisions in real-world applications.
Ques. What is the difference between Bayes Theorem and Total Probability Theorem? (3 Marks)
Ans. Bayes Theorem and Total Probability Theorem are both fundamental concepts in probability theory, but they have different purposes and applications. Here are the key differences between the two:
- Definition: Bayes Theorem is a mathematical formula that calculates the probability of an event based on prior knowledge, while the Total Probability Theorem calculates the overall probability of an event by summing the individual probabilities of all possible outcomes that could lead to that event.
- Purpose: Bayes Theorem is used to update the probabilities of events based on new information, while the Total Probability Theorem is used to calculate the overall probability of an event based on the probabilities of its sub-events.
- Application: Bayes Theorem is commonly used in machine learning, data analysis, and decision making, where it is used to update probabilities based on new data. The Total Probability Theorem is used to analyze complex probability distributions and to calculate the probability of an event based on multiple sources of information.
Ques. An event can occur in two ways only. If the difference of the probabilities of the two events is 20 %, what are the individual probabilities of the events? (5 Marks)
Ans. Let's call the probability of event 1 as P(1) and the probability of event 2 as P(2). From the problem statement, we know that the difference between the probabilities of the two events is 20%, or 0.20. We can express this as:
P(1) - P(2) = 0.20
We also know that the total probability of the sample space is equal to the sum of the probabilities of the individual events, or:
P(1) + P(2) = 1
To find the individual probabilities of the events, we can use these two equations to solve for one of the variables. For example, we can use the second equation to find P(1) in terms of P(2):
P(1) = 1 - P(2)
Then we can substitute this value into the first equation:
(1 - P(2)) - P(2) = 0.20
Simplifying this equation gives:
-P(2) = 0.20
Then by dividing both sides by -1 we get:
P(2) = 0.80
Therefore, P(2) = 0.80 or 80%
And P(1) = 1 - P(2) = 1 - 0.8 = 0.2 or 20%
So the individual probability of the two events is 20% and 80%.
Ques. A company has two factories, A and B. Factory A produces 70% of the company's products, and factory B produces the remaining 30%. What is the probability that a randomly chosen product was made in factory A? (2 Marks)
Ans. The sample space is all the products produced by the company, and the event of interest is a product made in factory A. The probability that a randomly chosen product was made in factory A is the percentage of products made in factory A divided by the total number of products: P(A) = 70/100 = 0.7
Ques. If 40% of boys opted for math and 60% of girls opted for math, then what is the probability that math is chosen if half of the class’s population is girls? (3 Marks)
Ans. Students can use this information to find the overall probability of choosing math, P(M), using the Total Probability Theorem. According to the theorem, we can find the overall probability of an event by finding the probability of the event happening given that each possible condition has occurred, and then multiplying that by the probability of the condition occurring, and adding all the results.
P(M) = P(M|boy) * P(boy) + P(M|girl) * P(girl)
Where P(boy) = 0.5 (since half of the class's population is girls) and P(girl) = 1 - P(boy) = 0.5 (since half of the class's population is girls)
So, P(M) = P(M|boy) * P(boy) + P(M|girl) * P(girl) = 0.40 * 0.5 + 0.60 * 0.5 = 0.50.4 + 0.50.6 = 0.20 + 0.30 = 0.50
Therefore, the probability that math is chosen if half of the class's population is girls is 50%.
Ques. A student is asked to solve 3 randomly chosen problems out of 5. What is the probability that the student will solve 2 problems correctly? (5 Marks)
Ans. The sample space is all the possible outcomes, which is {111, 112, 113, 121, 122, 123, 131, 132, 133, 141, 142, 143, 151, 152, 153, 211, 212, 213, 221, 222, 223, 231, 232, 233, 241, 242, 243, 251, 252, 253}.
The event of interest is solving 2 problems correctly. The probability of solving 2 problems correctly is the number of ways 2 problems can be solved correctly divided by the total number of possible outcomes: P(2 correct) = 10/60 = 1/6
Ques. An urn contains 4 white and 6 black balls. A ball is drawn at random, and then replaced and a second ball is drawn. What is the probability that both balls are white? (2 Marks)
Ans. The sample space is all the possible outcomes, which is {WW, WB, BW, BB}. The event of interest is drawing two white balls. The probability of drawing two white balls is the number of times two white balls can occur divided by the total number of possible outcomes: P(WW) = (4/10) * (4/10) = 0.16
Ques. A coin is flipped twice. What is the probability of getting two heads? (2 Marks)
Ans. The sample space is all the possible outcomes of the coin flip, which is {HH, HT, TH, TT}. The event we are interested in is getting two heads. The probability of getting two heads is the number of times two heads can occur divided by the total number of possible outcomes: P(HH) = ¼
Ques. A bag contains 5 red balls, 3 green balls and 2 blue balls. What is the probability of drawing a red ball? (2 Marks)
Ans. The sample space is all the balls in the bag, which is 5 red balls + 3 green balls + 2 blue balls = 10 total balls. The event we are interested in is drawing a red ball. The probability of drawing a red ball is the number of red balls in the bag divided by the total number of balls in the bag: P(red) = 5/10 = 0.5
Ques. A box contains 4 defective and 6 non-defective items. If an item is chosen at random, what is the probability that it is non-defective? (2 Marks)
Ans. The sample space is all the items in the box, which is 4 defective items + 6 non-defective items = 10 total items.
The event we are interested in is choosing a non-defective item.
The probability of choosing a non-defective item is the number of non-defective items in the box divided by the total number of items in the box: P(non-defective) = 6/10 = 0.6
Ques. A die is rolled twice. What is the probability of getting a 5 on the first roll and a 6 on the second roll? (3 Marks)
Ans. The sample space is all the possible outcomes of the die roll, which is {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6),...(6,6)}.
The event we are interested in is getting a 5 on the first roll and a 6 on the second roll.
The probability of getting a 5 on the first roll and a 6 on the second roll is the number of ways in which 5 and 6 can occur on first and second roll divided by the total number of possible outcomes: P((5,6)) = 1/36
Ques. A bag contains 2 red marbles, 3 blue marbles and 5 green marbles. If a marble is drawn at random, what is the probability that it is not blue? (2 Marks)
Ans. The sample space is all the marbles in the bag, which is 2 red marbles + 3 blue marbles + 5 green marbles = 10 total marbles.
The event we are interested in is not drawing a blue marble.
The probability of not drawing a blue marble is the number of non-blue marbles in the bag divided by the total number of marbles in the bag: P(not blue) = (2+5)/10 = 0.7
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