Understanding Mutually Exclusive Events in Probability Theory
Explore the concept of mutually exclusive events in probability theory, where the occurrence of one event excludes the other. Learn about conditional probability and find solutions to example scenarios involving event probabilities. Dive into the world of public health lectures and statistical analysis based on gender classifications.
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Presentation Transcript
Mutually Exclusive Events Two events are incompatible, disjoint or mutually exclusive when the occurrence of one precludes the occurrence of the other, i.e. they can not occur at the same time.
Example: Suppose a die is tossed. Then the events E = obtaining an even number and O = obtaining a one are mutually exclusive
Notice that Notice that ? ???????? ?? ???? ?????? ?? ??? = ?(?,?,?,?) =? ?= ? ? + ?(?). Note : For any two events ? and ?, ?(? ?) = ?(?) + ?(?) ?(? ?). If ? and ? are mutually exclusive then ?(? ?) = ?, and A and B cannot happen together, so that ?(? ?) = ?(?) + ?(?).
Conditional Probability The conditional probability of A given B is defined by:-? ? ?) = ?(? ?) ?(?) ???????? ???? ? ? ? Hence if A and B are any two events with probabilities greater than 0, then ? ? ? = ? ? ?)? ? .
Example# Example#: : A lecture on a topic of public health is held and 300 people attend. They are classified in the following way: Gender Gender Female Female Male Male Total Total Doctors Doctors 90 90 100 100 190 190 Nurses Nurses 90 90 20 20 110 110 Total Total 180 180 120 120 300 300 Find the following probabilities 1- ? ? ?????? ?? ?????? ; 2- ? ? ?????? ?? ?????? ; 3- ? ? ????? ?? ?????? ; 4- ? ? ???? ?? ?????? ; 5- ? ? ?????? ????? ?? ?????? ; 6- ? ? ???? ?????? ?? ?????? ;
1 1- - The number of doctors is The number of doctors is 190 number of people is number of people is 300 190 and the total and the total 300, so , so ? ?????? =??? ??? 2 2- - ? ?????? =??? ??? 3 3- - ? ???? =??? ??? 4 4- - ? ????? =??? ??? female nurses, therefore ?? ??? 5 5- - There are There are 90 ? ?????? ????? = 90 female nurses, therefore 6 6- - ? ???? ?????? = ? ???? ?????? =??? ???
Now suppose you are given the information that a female is chosen and you wish to find the probability that she is a nurse. This is what we call conditional probability. We want the probability that the person chosen is a nurse, subject to the condition that we know she is female. The notation used for this is: ? ????? ?????? Read this as the probability of the person chosen being a nurse, given that she is female . Since there are 180 females and of these 90 are nurses, the required probability is ?? ???=? ?. ?? ??? We can see that ? ????? ?????? = ?? ??? ??? ??? = ?(????? ??????) ?(??????) =
Independent Events Incidents A or B are independent incidents If the occurrence of one of them or not Its occurrence does not affect the occurrence of the other . For example, when throwing a coin twice in a row the result of the second throw is not affected by the result of the first. Two events A and B are said to be independent if and only if ? ? ? = ? ? , that is, when the conditional probability of A given B is the same as the probability of A. Example ## :In the problem of nurses and doctors given on Example # , define A to be the event a nurse is chosen and B to be the event a female is chosen . Are the events A and B independent?
Solution: Solution: ?? ???=? ? ? ? = ? ? ? =??? ? ? ? ??? So A and B are not independent.
Note: From the definition of conditional probability we have ?(? ?) = ?(? | ?).?(?) Now if ? and ? are independent, then ?(? | ?) = ?(?),?? ?(? ?) = ?(?).?(?). When two events are independent, the chance that both will happen is found by multiplying their individual chances. This gives us a simple way of checking whether or not events are independent: ? and ? are independent events if and only if ?(? ?) = ?(?).?(?). Note: It is possible to define independence in this way without referring to conditional probability.
: . O P( P(??) ) P( P(??) ) O Sol. Sol. S S1 1={ H, T } ={ H, T } , , s s2 2={ ={ H, T H, T } } , , )= P( P(??) ). . P( P(??) )= =? ? P(A P(A ?)= P(A).P(B) )= P(A).P(B) ? ?= ? ? P( P(????)= :
Complement Let ? be an event. The complement of ?, denoted by ??, corresponds to the event that ? does not occur. By definition, we have ? ?? = (the empty set) and ? ??= ?. Here, the intersection operation corresponds to and ; and the union operation corresponds to or . Since ? = ? ? = ? ? ??= ? ? + ? ??, ?? ???? ?(??) = ? ?(?) .
Example: Roll of a Die =? ? ? ??? ??????? ?? ??? ? ? = ? ? ? ? =? ?. ?
