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The addition rule for probabilities consists of two separate rules or formulas. The first rule takes into account the occurrence of two events that are incompatible with one another, while the second rule takes into account the occurrence of two events that are compatible with one another.
The fact that the two events in question are not mutually exclusive indicates that there is some degree of overlap between them. The formula accounts for this overlap by deducting the probability of the overlap, denoted by P(Y and Z), from the total probability of Y and Z.
In principle, the first version of the rule can be understood as a particular instance of the second form.
Using the definition of probability, one can quickly and readily determine the likelihood that a certain event will take place. On the other hand, the definition alone cannot be used to calculate the likelihood that at least one of the events listed will take place. These kinds of issues can be resolved by applying a theorem that goes by the name “Addition theorem.” The following is a statement of the “Addition theorem,” followed by its proof and different applications of the theorem.
Mutually Exclusive Events:
If there is no link between two or more occurrences, we say that they are mutually exclusive. This means that they cannot occur simultaneously. To put it another way, if the happening of one of the events makes it impossible for the others to take place, then we say that the occurrences are mutually exclusive.
Example:
When two coins are tossed, the possibility of receiving two heads, A, and the possibility of getting two tails, B, are incompatible with one another.
Because A = {HH}; B = {TT}.
Mutually Exhaustive Events:
When both of a set of occurrences are impossible to take place, we say that those events are mutually exhaustive. This means that none of those events can take place. That is to say, undoubtedly, one of those occurrences will take place.
If both A and B completely exhaust each other, then the probability of them coming together is one hundred percent.
i.e. P(AUB)=1.
Example:
When a coin is tossed, the possibility of landing on one’s head and the possibility of landing on one’s tail are both equally unlikely.
Addition Theorem on Probability:
If two occurrences are denoted by the letters A and B, then the probability that at least one of the events will occur can be calculated as follows: P(AUB) = P(A) + P(B) – P(AB).
Proof:
Considering that occurrences are nothing more than sets,
The following is derived from set theory:
n(AUB) = n(A) + n(B)- n(A∩B).
When the above equation is divided 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)
Therefore, according to the accepted definition of probability,
P(AUB) is equal to P(A) + P(B) – P(AB).
Example:
If George and James, two students, have a probability of solving a problem that is 1/2 and 1/3, respectively, then what is the likelihood that the problem will be solved?
Solution:
Let’s say that A and B represent the odds that George and James will be able to solve the problem, respectively.
Therefore, P(A) equals 1/2 and P(B) equals 1/3.
If at least one of them is able to find a solution to the problem, then it can be considered solved.
Therefore, we need to locate P. (AUB).
According to an extension of the addition theorem for probability, we obtain
P(AUB) is equal to P(A) plus P(B) minus P(AB).
P(AUB) = 1/2 +.1/3 – 1/2 * 1/3 = 1/2 +1/3-1/6 = (3+2-1)/6 = 4/6 = 2/3
Note:
If A and B are any two events that cannot occur simultaneously, then the probability of P(A-B) is zero.
Therefore, P(AUB) = P(A) + P (B).
Conclusion
The addition rule for probabilities is comprised of two separate rules or formulas. The first rule takes into account the occurrence of two events that are incompatible with one another, while the second rule takes into account the occurrence of two events that are compatible with one another.
The fact that the two events in question are not mutually exclusive indicates that there is some degree of overlap between them. The formula accounts for this overlap by deducting the probability of the overlap, denoted by P(Y and Z), from the total probability of Y and Z.
