Tips And Tricks And Shortcut For Venn Diagrams

Tips and Tricks and Shortcuts for Venn Diagrams:-

Venn Diagrams are a schematic and identical way of representing the elements of a set or a group.

In mathematical language, it represents the intersection of the two groups.

In the above diagram, we see that there are three groups or sets called ‘A’ ,’B’, and ‘C’. These three sets could represent any given collection of people.

For example-
A contains all the people that like cake. Set B represents all the people who like ice cake and set C represents all the people who like chips. Then the region marked as AB represents all the people who like both cake and ice cake. The region marked BC represents all the people who like both ice cake and chips. Similarly, the region AC represents all the people who like cake and ice cake.

If there are only two elements

n ( A ∪ B) = n(A ) + n ( B ) – n ( A∩ B)

If there are three elements

n (A ∪ B ∪ C) = n(A ) + n ( B ) + n (C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) + n (A ∩ B ∩ C)

Question 1

In a class of 260 seniors, 93 study Spanish, 95 study Chemistry, 165 study Mathematics, 18 study Spanish and Chemistry, 75 study Chemistry and Math, 20 study Math and Spanish and 15 study all three subjects. Make a Venn diagram to illustrate the data and then find the probability that a student selected at random studies:

Options

(a) 0.121
(b) 0.0192
(c) 0.269
(d)0.387

Explanations

P(S∩M’∩C’) = $\frac{70}{260} \$ = $\frac{7}{26} \$ = 0.269

maths and chemistry but not spanish

P(MCS)= $\frac{60}{230} \$ = $\frac{3}{13} \$ = 0.231

None of these subject

P(MCS) = $\frac{5}{260} \$ = $\frac{1}{52} \$

Spanish, given that he/she studies Math

P(S|M) =$\frac{P(S∩M’)}{P(M)} \$  = $\frac{20}{260} \$ $\frac \$ $\frac{165}{260} \$ = $\frac{4}{33} \$ = 0.121

Question 2

Given P(A∩B) = 0.4, P(A∩B’) = 0.2 and P(A’∩B’) = 0.3, find P(B) and P(A|B).

Options

(a) 0.8
(b) 0.6
(c) 0.4
(d) 0.2

Explanations

The missing value, P(B∩A’), must be 0.1 in the order for the total of the probabilities in the sample space equal to 1. Thus , P(B) = 0.5
P(A|B) = $\frac{P(A∩B)}{P(B)} \$ = $\frac{P(0.4)}{P(0.5)} \$ = 0.8

Read Also: How to solve Venn Diagrams question