Errata: Are you a Cylon in Battlestar Galactica?

There's an error in my previous analysis of the probability of me being a Cylon if you've gotten to look at one of my loyalty cards randomly and it was a Human card. We seemingly innocently transformed the previous scenario into the condition that I have one or more human cards, which are not quite the same thing. While it is true that I have one or more human cards, you have additional information: that a randomly selected card was human. Actually, even if you made the statement about having one or more Cylon cards, we'd still be in the same situation, as all Cylon detection abilities either allow you to look at all Loyalty cards or one randomly (see this handy chart: https://boardgamegeek.com/filepage/41329/crisis-card-chart). If there were an ability that allowed the target to select the card to be observed, the analysis would hold.

To see what's happening here, let's look at a three player game to make the number of cases simpler. You, Chelsea, and I are playing and you get to look at one of our loyalty cards. All of us have two cards, you know you're human, so there's one Cylon card among the four loyalty cards that Chelsea and I have. I could either have the Cylon card or not, without loss of generality we can say that it's my "first'' card. You randomly get one of my cards. Table 1 shows all the cases, which occur with equal probability.

\begin{align*} \begin{array}{c|c|c|c|c} \text{Scenario #} & \text{Loyalty card 1 } & \text{Loyal card 2} & \text{Card # observed} & \text{Card observed} \\ \hline 1 & \text{Cylon} & \text{Human} & 1 & \text{Cylon} \\ 2 & \text{Cylon} & \text{Human} & 2 & \text{Human} \\ 3 & \text{Human} & \text{Human} & 1 & \text{Human} \\ 4 & \text{Human} & \text{Human} & 2 & \text{Human} \\ \end{array} \end{align*}

Table 1: Cylon Detection in Battlestar Galactica

As we can see here, the probability of me being a Cylon given that you randomly looked at one of my cards, which was human, is $1/3$. There are three cases where you see a human card, one of which involves me having a Cylon card.

Let's go back to the five player case. The differentiating factor is that the probability of me getting one Cylon card is not $1/2$, but instead $24/56=3/7\approx0.429$ as we computed before, and the probability of me getting two human cards is $30/56=15/28\approx0.536$. We don't care about the two Cylon card case, as we're only looking at the scenarios where you see a human card. The probability of that happening is as follows.

\begin{align} P(\text{see human}) &= \underbrace{0.5}_\text{Chance to see each of my cards.} \cdot P(\text{1 Cylon}) + P(\text{human})\\ &=0.5 \cdot \frac{24}{56} + \frac{30}{56} \\ &=\frac{12+30}{56} \\ &=\frac{3}{4}\\ &=0.75 \end{align}

The probability that I am Cylon, given that you see a human card from me uses this probability in the denominator.

\begin{align} P(\text{Cylon} | \text{see human}) &= \frac{P(\text{1 Cylon} \cap \text{see human})}{P(\text{see human})} \\ &= \frac{0.5 \cdot P(\text{1 Cylon})}{P(\text{see human})} \\ &= \frac{0.5 \cdot \frac{24}{56}}{\frac{3}{4}}\\ &=\frac{12}{56} \cdot \frac{4}{3}\\ &=\frac{4}{14}\\ &=\frac{2}{7} \end{align}

This is the same probability as if we saw that the first loyalty card was human before the sleeper agent phase.