What Is the Proof of Addition Theorem by Using Probability?
Let us first understand the term, “probability”. The probability of an event occurring is defined by probability. There are numerous circumstances in which we must anticipate the result of an event in real life. We may be uncertain or certain about the outcome of an event. In such circumstances, we say that the event has a chance of happening or not happening. Probability has several uses in games, in business to generate probability-based forecasts, and in this emerging area of artificial intelligence. The ratio of the number of favourable outcomes to the total number of outcomes of an event is known as probability. The number of positive outcomes can be represented as x in an experiment with n number of outcomes.
The following is the formula for calculating the probability of an occurrence. Probability (Event) = Favourable Outcomes/Total Outcomes = x/n. To better grasp probability, let’s look at a basic example. If we need to check whether rain is falling or not. “Yes” or “No” is the response to this question. There is a chance that it will rain or not rain. We can use probability in this situation. Probability is used to predict the results of coin tosses, dice rolls, and card draws from a deck of playing cards. Theoretical and experimental probabilities are the two types of probabilities.
Basic Theorems on Probability
The following probability theorems are useful in understanding probability applications and performing various probability calculations.
Theorem 1 – The chance of an event occurring and the probability of an event not occurring add up to one. P(A)+P(¯A) =1
Theorem 2 – The probability of an impossible event or an event that does not occur is always equal to zero.P(ϕ)=0.
Theorem 3 – A certain event’s probability is always equal to one.P(A) = 1.
Theorem 4 – Any event’s chance of occurring is constantly between 0 and 1.0 < P(A) < 1.
Theorem 5 – If two events A and B occur, we may use the formula for the union of two sets to obtain the formula for the likelihood of event A or event B occurring, which is as follows.P(A∪B) =P(A)+P(B)−P(A∩B). We also get P (A U B) = P(A) + P(B) for two mutually exclusive occurrences A and B. (B)
Addition Theorem
Using the definition of probability, you may quickly determine the likelihood of an event occurring. However, the definition alone cannot be used to determine the likelihood of at least one of the events occurring. The “Addition theorem” is a theorem that addresses these kinds of issues. The following contains the formulation and demonstration of the “Addition theorem,” as well as its application in various situations.
Mutually exclusive events: If two or more occurrences have no elements in common, they are said to be mutually exclusive. i.e., if the occurrence of one event prohibits the occurrence of the others, they are said to be mutually exclusive. When two coins are tossed, the events of obtaining two heads, A, and receiving two tails, B, are mutually exclusive.Because A = {HH}; B = {TT}.
Mutually exhaustive events: If there is a confidence that at least one of the two events will occur, they are said to be mutually exhaustive. That is to say, one of those occurrences will undoubtedly occur. If A and B are mutually exhaustive, their union has a probability of one. P(AUB)=1, for example.For example, when a coin is tossed, the events of receiving a head and receiving a tail are mutually exhaustive.
Addition theorem on probability: If A and B are any two occurrences, then P(AUB) = P(A) + P(B)- P(AB) is the probability of at least one of them occurring. Proof: Because events are nothing more than sets, we have set theory to thank for this, giving, n(AUB) = n(A) + n(B)- n(A∩B). By dividing the previous equation by n(S), we get (where S is the sample space). n(AUB)/ n(S) = n(A)/ n(S) + n(B)/ n(S)- n(A∩B)/ n(S). Then, according to the probability definition, P(AUB) = P(A) + P(B)- P(A∩B).
Example
If the likelihood of two students, Sam and Jack, answering a problem is 1/2 and 1/3, what is the probability of the issue being solved?
Solution: Let A and B represent the chances of Sam and Jack solving the issue, respectively. Then P(A) equals 1/2 and P(B) equals 1/3. If at least one of these solves the problem, it will be solved. As a result, we must locate P. (AUB). P(AUB) = P(A) + P(B)- P(AB) is the probability addition theorem. P(AUB) = P(A) + P(B)- P(A∩B). P(AUB) = 1/2 +. 1/3 – 1/2 * 1/3 = 1/2 +1/3-1/6 = (3+2-1)/6 = 4/6 = 2/3.
It’s worth noting that if A and B are mutually exclusive occurrences, then P(AB)=0. Then P(AUB) = P(A)+P(B).
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