Are considered mutually non-exclusive events to all those events that have the ability to occur simultaneously in an experiment. The occurrence of one of them does not imply the non-occurrence of the other.
Unlike their logical counterpart, mutually exclusive events, the intersection between these elements is different from the void. This is:
A ∩ B = B ∩ A ≠ ∅
Because the possibility of simultaneity between results is handled, mutually non-exclusive events require more than one iteration to cover probabilistic studies..
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In probability, two types of eventualities are handled; The occurrence and non-occurrence of the event. Where the binary quantitative values are 0 and 1. Complementary events are part of relationships between events, based on their characteristics and particularities that can differentiate or relate them to each other..
In this way, the probabilistic values run through the interval [0, 1] varying their parameters of occurrence according to the factor sought in the experimentation..
Two mutually non-exclusive events cannot be complementary. Because there must be a set formed by the intersection of both, whose elements are different from the void. Which does not meet the definition of complement.
They are possibilities and events resulting from experimentation, capable of offering results in each of their iterations. The events generate the data to be recorded as elements of sets and sub-sets, the trends in these data are the subject of study for probability..
Let A and B be two mutually non-exclusive events belonging to the sample space S.
A ∩ B ≠ ∅ and the probability of occurrence of their intersection is P [A ∩ B]
P [A U B] = P [A] + P [B] - P [A ∩ B]; This is the probability that one event or another will occur. Due to the existence of common elements, the intersection must be subtracted so as not to add twice.
There are tools in set theory that greatly facilitate working with mutually non-exclusive events..
The Venn diagram between them defines the sample space as the universe set. Defining within it each set and subset. It is very intuitive to find the intersections, unions and complements that are required in the study.
A juice vendor decides to end his day and give the rest of his merchandise to each passerby. For this, serve in 15 glasses all the juice that was not sold and put a lid on them. He leaves them on the counter for each person to take the one they prefer.
It is known that the seller was able to fill
Define the probability that the following mutually exclusive events occur when drinking a glass:
The second property is used; P [A U B] = P [A] + P [B] - P [A ∩ B]
Where, depending on the case, we will define the sets A and B
1-For the first case, the groups are defined as follows:
A: be citric = n1, n2, n3, n4, n5, n6, l1, l2, l3
B: be orange = n1, n2, n3, n4, n5, n6, m1, m2, m3
A ∩ B: n1, n2, n3, n4, n5, n6
To define the probability of an event we use the following formula:
Specific case / Possible cases
P [A] = 9/15
P [B] = 9/15
P [A ∩ B] = 6/15
P [A U B] = (9/15) + (9/15) - (6/15) = 12/15
When this result is multiplied by 100, the percentage of possibility that this event has is obtained.
(12/15) x 100% = 80%
2-For the second case, the groups are defined
A: be citrus = n1, n2, n3, n4, n5, n6, l1, l2, l3
B: be green = l1, l2, l3
A ∩ B: l1, l2, l3
P [A] = 9/15
P [B] = 3/15
P [A ∩ B] = 3/15
P [A U B] = (9/15) + (3/15) - (3/15) = 9/15
(9/15) x 100% = 60%
3-For the third case, proceed the same
A: be fruit = n1, n2, n3, n4, n5, n6, l1, l2, l3, m1, m2, m3, s1, s2, s3
B: be green = l1, l2, l3
A ∩ B: l1, l2, l3
P [A] = 15/15
P [B] = 3/15
P [A ∩ B] = 3/15
P [A U B] = (15/15) + (3/15) - (3/15) = 15/15
(15/15) x 100% = 100%
In this case, the condition "Let it be fruit" includes the entire sample space, making the probability of 1.
4- For the third case, proceed the same
A: not citrus = m1, m2, m3, s1, s2, s3
B: be orange = n1, n2, n3, n4, n5, n6, m1, m2, m3
A ∩ B: m1, m2, m3
P [A] = 6/15
P [B] = 9/15
P [A ∩ B] = 3/15
P [A U B] = (6/15) + (9/15) - (3/15) = 12/15
(12/15) x 80% = 80%
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