# Binomial probabilities in games when players can lie

An oracle can be asked a binary question. If the real answer is true, the oracle says true with probability tt (and false with probability tf=1-tt). If the real answer is false, the oracle says false with probability ff (and true with probability_ft=1-ff_). So the oracle is defined as P(q) = { tt, ff}.

When the oracle is asked the same question n times and answers true at least m times, what is the probability that the real answer is true?

• By and large, you're better off posting these sorts of Probability questions over at math.stackexchange.com Jul 5 '14 at 17:05

Let $R$ represent if the real answer is true ($R=T$) or not ($R=F$). Let $N$ ($<n$) be the number of times the oracle says true. The required quantity is $\text{P}(R=T|N\ge m)$. Use Bayes theorem:
$$\text{P}(R=T|N\ge m) = \frac{\text{P}(N\ge m|R=T)\text{P}(R=T)}{\text{P}(N \ge m)}$$
$$\left( N|R=T \right) \sim \text{Bin}(n, tt)$$
A similar expression for $N|R=F$ can be obtained, allowing the calculation of the probabilities involving $N$ in the question.
No information regarding $\text{P}(R=T)$ is given, so a prior probability of $\frac{1}{2}$ might be assumed.