If the two events are mutually exclusive, then the circles representing each event do not overlap.A two-event Venn diagram describes the relationship between two events in the following ways:.probability or relative proportion of an event.Each of the divided regions in a Venn diagram can contain one of the following data:.In general, the conditional probability of event □ given another event □ can be computed using a Venn diagram and following similar steps. In our previous example, we computed a conditional probability using a Venn diagram. Hence, the probability that a randomly selected student failed physics given that he passed chemistry is equal to 0.1. s t u d e n t s w h o f a i l e d p h y s i c s s t u d e n t s w h o p a s s e d c h e m i s t r y Substituting these values into the equation above, we obtain s t u d e n t s w h o f a i l e d p h y s i c s a n d p a s s e d c h e m i s t r y s t u d e n t s w h o p a s s e d c h e m i s t r y s t u d e n t s w h o f a i l e d p h y s i c s s t u d e n t s w h o p a s s e d c h e m i s t r y s t u d e n t s w h o f a i l e d p h y s i c s a n d p a s s e d c h e m i s t r y s t u d e n t s w h o p a s s e d c h e m i s t r yįrom the green- and purple-shaded regions of the Venn diagram above, we can obtain In our example, this formula can be written as Let us check this answer using the formula for the conditional probability for event □ given event □: P e r c e n t a g e o f s t u d e n t s w h o f a i l e d p h y s i c s b u t p a s s e d c h e m i s t r y p e r c e n t a g e o f s t u d e n t s w h o p a s s e d c h e m i s t r y = 6 % 6 0 % = 0. So, the probability of randomly selecting a student who failed physics from the group of students who passed chemistry is given by Among the students who passed chemistry, the percentage of students who failed physics is found in the green-shaded region, which is 6 %. The percentage of students who passed chemistry is given by summing the percentages within the purple region, which leads to 6 % + 5 4 % = 6 0 %. In other words, our sample space (or universal set) is the purple-shaded region above, not the rectangle. This means that we are selecting a student from the group of students who passed chemistry rather than from the entire class. Notice that we are given that the student passed chemistry. In our next example, we will organize the given data using a Venn diagram and use it to compute the probability of an event given a condition. We can also use a Venn diagram to compute conditional probabilities. Hence, the probability that a randomly selected number between 1 and 12 is not a multiple of 3 is 8 1 2, which can be simplified to 2 3. We can see that there are 8 numbers that belong to this case. Hence, the probability of this event is equal to 0.įrom the Venn diagram, the numbers between 1 and 12 that are not a multiple of 3 lie outside of the circle. In other words, there is no number between 1 and 12 which is both a factor of 20 and a multiple of 3. We note that the two events are mutually exclusive since the circles in the Venn diagram do not overlap. Hence, the probability of selecting a number between 1 and 12 that is a factor of 20 is 5 1 2. We can see from the Venn diagram that there are 5 different factors of 20 between 1 and 12. f a c t o r o f n u m b e r o f f a c t o r s o f Since we are randomly selecting a number between 1 and 12, the probability of an event can be obtained from dividing the size of the event by the size of the sample space, which is 12. In other words, the sample space (or the universal set) is given by In the given Venn diagram, two events are represented as circles: “Factor of 20” and “Multiple of 3.” The sample space of this Venn diagram is the set of integers between 1 and 12. What is the probability of selecting a number that is not a multiple of 3? Give your answer as a fraction in its simplest form.What is the probability of selecting a number that is a factor of 20 and a multiple of 3? Give your answer as a fraction in its simplest form.What is the probability of selecting a number that is a factor of 20? Give your answer as a fraction in its simplest form.
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