Intersection of Probabilities and Bayes Theorem

Question

It is known that 70% of women and 60% of men have voted in a poll, in a village where 500 women and 400 men live. If only 80% of the inhabitants tells the truth, what is the probability that a person who says "I did vote" is telling the truth?

I've broken this problem down into the events:

V = The person voted

T = The person tells the truth

I know as well that I'm looking for $P(T|V)$. I know in order to calculate this, by Bayes Theorem, I need to calculate $P(V|T)$.

I have that $P(V) = \frac{59}{90}$ and $P(T) = \frac{8}{10}$. My problem is that I don't know that I have enough information to calculate $P(V \cap T)$ to be able to continue and get a concrete probability for $P(T|V)$.

Answer 1

$350+240=590$ of $500+400=900$ voted. $472$ of those who voted would say they voted while $62$ of those who didn't vote would say they voted. Thus, $\frac{472}{472+62}=\frac{236}{267}$ of those who said they voted would actually have voted.

Answer 2

You are thinking it the wrong way:

The person who said I did vote has either vote and telling the truth or not voted and lying.

So:

let B: the person says that voted

let T: the person is telling the truth

let V: the person voted

$P(B) = P(V \cap T) + P(notV \cap notT)$

so B: voted and tells the truth plus not voted and lie

$P(T|B) = \frac{P(T \cap B)}{P(B)}$

$P(T|B) = \frac{P(T \cap V)}{P(V \cap T) + P(notV \cap notT)}$

$P(V) = \frac{0.7 \times 500 +0.6 \times 400}{900} = 590/900$

$P(T) = 0.8 $

so

$P(T|B) = \frac{\frac{590}{900}\times 0.8}{\frac{590}{900}\times 0.8 + (1-\frac{590}{900}) \times 0.2}$

Answer 3

Refer to the tree diagram:

$\hspace{2cm}$

The required probability is: $$\frac{\text{n(people who claim to have voted and had voted indeed)}}{\text{n(people who claim to have voted, but they may or may not have voted)}}=\\ \frac{WVT+MVT}{(WVT+MVT)+(WN'L+MV'L)}=\\ \frac{500\cdot 0.7\cdot 0.8+400\cdot 0.6\cdot 0.8}{(500\cdot 0.7\cdot 0.8+400\cdot 0.6\cdot 0.8)+(500\cdot 0.3\cdot 0.2+400\cdot 0.4\cdot 0.2)}=\\ \frac{472}{472+62}=\frac{236}{267}\approx 0.88.$$ Note:

1) WVT - women voted and telling truth (they did vote);

2) WV'L - women did not vote, but lying (they said they had voted);

3) MVT - men voted and telling truth (they did vote);

4) MV'L - men did not vote, but lying (they said they had voted).