I don't think you are correct with your statement of "The probability that all of the other women are lying is .1%".
This is a prior probability/Bayes Theorem situation.
If we didn't know that the first woman lied, then, yes, we could calculate that the probability of all three women lying is 10% x 10% x 10% = 0.1% (this assumes that each woman's probability is independent of the others).
But we have new information - the first woman lied (at least, that's how things look). So the new question is:
Given that the first woman lied, what is the probability that all the other women lied?
The Bayes Theorem states
P(A | B) = P(A) × P(B | A) / P(B)
where:
P(A | B)= the conditional probability of all women lying given the first woman lied
P(A) = the prior probability of all women lying (0.1%)
P(B | A) = the conditional probability that the first woman lied given that all the women lied
P(B) = the probability that the first woman lied (10%)
We don't really know P(B | A) but we could guesstimate. Maybe Bauer just attracts women who tend to lie in order to get money? We could set it at 90%.
The calculation would then be:
P(A | B) = 0.1% x 90% / 10% = 0.9%
Now you can go 2 ways with this result.
One way is that 0.9% is still a pretty small probability that all women are lying.
The other way is that the fact that the first woman lied means that it's now 9 times more likely that all of the other women are lying.
I'm just a lowly mathematician. Feel free to critique my work.