Refer to this post for all direct formulas that can be used for solving venn diagram problems
https://www.beatthegmat.com/formulas-for ... 16479.html
Let the total number of households be 100
All houses do not have either tubelight or bulb or fan.
So we have,
19 houses do not have just one of these.
67houses do not have tubelights.
83houses do not have bulbs.
73houses do not have fans.
(i) Number of ppl lacing all three utilities
Lets use our regular formula
P(TuBuF) = P(T) + P(B) + P(F) - {P(TnB) + P(BnF) + P(FnT)} + P(TnBnF)
Let P(TnB) + P(BnF) + P(FnT) = X
We Get,
P(TuBuF) = P(T) + P(B) + P(F) - {X} + P(TnBnF) ....(1)
keep that aside for now.
Now, it is given that 19 houses do no have just one of these. Which means we are looking at the higlighted area of the venn diagram i.e. the number of ppl in exactly one set. Understand this part and the rest is just formula substitution.
No of persons in exactly one set = P(T) + P(B) + P(F) - 2P(TnB) - 2P(BnF) - 2P(FnT) + 3P(TnBnF)
(since P(TnB) + P(BnF) + P(FnT) = X)
No of persons in exactly one set = P(T) + P(B) + P(F) - 2{X} + 3P(TnBnF)
19=223-2{X}+3P(TnBnF)
2{X}=204+3P(TnBnF)
{X}={204+3P(TnBnF)}/2
Substitute this value of X in (1)
100=223-{204+3P(TnBnF)}/2+P(TnBnF)
-123={-204-3P(TnBnF)}/2+P(TnBnF)
-246=-204-3P(TnBnF)+2P(TnBnF)
-246=-204-P(TnBnF)
P(TnBnF)=42=Number of ppl lacking all three utilities
Substituting this value back in in statement (1)
100=223-{X}+42
{X}=123+42=165
(ii)Number of ppl lacking exactly two utilities
No of persons in exactly two of the sets = P(TnB) + P(BnF) + P(FnT) - 3P(TnBnF)={X}-3P(TnBnF)
=165-126=39
Refer to the link above for all set related formulas. Its just simple substitution.
