Question

[Discrete math/sets:]

universe = 151x
A = 68x
B = 61x
C = 52x

A intersect B = 16x
A intersect C = 25x
B intersect C = 19x

where x is the number of people who watch the TV show that any set represents (universe is the number of people being polled)

How many persons watched all three shows?

Answer 1

Does everyone have to be in either A, B or C (or a combination of these).. like is there a possibility that there are people outside of (A or B or C) ?

Answer 2

I wouldn't know. Problem only states that 151 were polled, and those were the findings.

Answer 3

Well I know you can use this rule:
\(\begin{align}P(A\cup B \cup C)&=P(A)+P(B)+P(C)\\&~~~-P(A\cap B)-P(A \cap C)-P(B \cap C)+P(A \cap B \cap C)\end{align} \)

And if everyone has to be part of at least A, B or C, then you could assume \(A \cup B \cup C\) to be the universe. Then you just solve for \(P(A \cap B \cap C)\)

Answer 4

Yeah how do I solve for it though... that's my original question... The first term = 121 + whatever the answer is for my original question. Right? So... how do I solve for something to solve what I'm trying to solve.. for... ugh.

Answer 5

is that how the whole question was posted in its entirety?

Answer 6

Yeah:
'A TV poll of 151 persons found that 68 watched "Law and Disorder"; 61 watched "25"; 52 watched "The Tenors"; 16 watched both "Law and Disorder" and "25"; 25 watched both "Law and Disorder" and "The Tenors"; 19 watched both "25" and "The Tenors". How many persons watched all three shows?'

Answer 7

Ok. Well there is no information about the number of people who did not watch any of the shows... I guess you could assume this is 0? So, this would make your universe to consist of \(A \cup B \cup C\). I'm just going to relabel though:
L = Law and Disorder
T = 25
E = The Tenors

Instead of going though a probability argument, you can equally use the notation \(n(T \cap E)=19\) for saying the the number of people.

So:
\(\begin{align}n(L \cap T \cap E)&=n(L\cup T \cup E)-n(L)-n(T)-n(E)\\&~~~+n(L\cap T)+n(L \cap E)+n(T \cap E)\\ &=151-68-61-52+16+25+19\end{align}\)