You are waiting for your flight to Seattle, and to pass the time you call 3 friends in Seattle. You independently ask each one if it is raining.
All 3 of your friends say “Yes, it is raining.”
But each friend lies with probability $1/3$ and tells the truth with probability $2/3$ (independently).
Can you solve for probability it is actually raining today in Seattle?
Solution A (Bayesian approach)
Lets using $P(rain)$ to denote the prior probability of raining in Seattle, $P(dry)=1-P(rain)$ is the prior probability of Seattle stays dry, and $P(y,y,y)$ is the probability of getting 3 “Yes” answers from your friends.
The probability of rain given 3 “Yes” answers is
$$ \begin{align*} P(rain|y,y,y)&=\frac{P(rain)\cap P(y,y,y)}{P(y,y,y)}\\ &=\frac{P(y,y,y|rain)P(rain)}{P(y,y,y|rain)P(rain)+P(y,y,y|dry)P(dry)}\\ \end{align*} $$
Set $P(rain)=p$, then $P(dry)=1-p$, the above formula becomes:
$$ \begin{align*}\frac{(2/3)^3\cdot p}{(2/3)^3\cdot p+(1/3)^3\cdot (1-p)}&=\frac{8p/27}{8p/27+(1-p)/27}\\&=\frac{8p}{7p+1} \end{align*} $$
Therefore the probability of raining today in Seattle is $8p/(7p+1)$.
Solution B (Conditional probability with Logic)
Since you don’t have access to the actual historical raining data in Seattle (prior probability), you need to give the best estimation based on information you’ve given (currently the sole information source are your friends’ answers).
There’s $2/3\cdot2/3\cdot2/3=8/27$ chance they’re all telling the truth.
And $1/3\cdot1/3\cdot1/3=1/27$ chance they’re all lying.
By logic, they’re either all telling the truth or all lying, there’s impossible 2 of them are lying and 1 of them telling the truth (Contradiction).
From your perspective, the answer from your friends (Yes or No) and the weather status today in Seattle (Rain or Dry) are perfectly dependent: If they’re all telling the truth, then it’s definitely raining, and it’s dry if they’re all lying.
So the probability of It’s raining and they’re all telling the truth is equal to the probability of they’re all telling the truth, which is $\frac{8/27}{8/27+1/27}=8/9$.