"Why does he reach for this starship?"

"To protect us. It's an Autobot thing You wouldn't understand, Megatron."

—Bumblebee and 1/2 of Megatron, Robots in Disguise #26

The analysis of Raider Vanguard is a good reason to start including the effect of Focus in our charts on the probability of flipping white icons (see White Icons & Abilities). We start with Focus 1, and consider the probability of flipping at least one white icon. We're considering here the effect of Focus 1 both when we don't have any Tough, as well as in conjunction with low counts of it. These are the quantities plotted in these three panels (click on the figure to change the value of Tough). They are derived under the assumption that the top card of our deck is always scrapped when not white, i.e. the effect of Focus 1, given the goal we're trying to achieve with Vanguard. These probabilities are plotted as a function of the number of white cards in our deck, as they only depend on it.

Technical note: Each point in these figures is generated by flipping cards from the top of 200'000 simulations of its corresponding 40-card decks. Every number referred to as a "probability" is actually a normalized occurrence number.

As expected, as our Tough count increases, the boost provided by Focus 1 becomes less and less impactful (difference between the dotted and the dashed line in each panel). Therefore, Focus 1 provides its greatest contribution when we don't have any Tough. But, in that case, its boost can be as high as +15%, if we're playing between 3 and 5 playsets of white cards. Interestingly enough, this range includes the optimal number of white cards (about 10, see Variance, Part 2) that we would play to simultaneously optimize both our attack and defense bonuses when not relying on Bold and Tough.

A simple rule of thumb is that with 10 white cards, Focus 1 and no Tough, our chances of benefitting from Vanguard's defensive ability are as high as 60%. Without Focus or Tough, and with the same number of white cards, it would be a less exciting 45%.

Vanguard's Theorem

A second look at the the previous charts shows an unexpected similarity between the probability of activating Vanguard's ability with Focus 1 and no Tough, and the probability corresponding to Tough 1 and no Focus, i.e. like Focus 1 and Tough 1 were, individually taken, providing the same boost to our probability of flipping at least one white icon. The same is true when comparing Focus 1 plus Tough 1 with Tough 2 but no Focus. This apparent coincidence happens regardless of the number of white cards we play... In fact, it's not a coincidence, but the consequence of a much more general result, stating that these probabilities are exactly the same. Here on Kimia, we sometimes refer to this result as Vanguard's Theorem.

Vanguard's Theorem: The probability of flipping at least 1 white icon with Focus X is equal to the probability of flipping at least 1 white icon with Bold/Tough X.

Proof: The probability of of flipping at least 1 white icon is equal to 1 minus the probability of not flipping any white. Therefore, it's enough to evaluate these probabilities of not flipping white −first with both Bold/Tough X, and then with Focus X− and show that they are the same. Let's call PW the probability of flipping a white card. With Bold/Tough X, we're flipping 2 + X cards, and the probability that none of them is white is (1 − PW)2+X. With Focus X, we first look at the top X cards. None of them is white with probability (1 − PW)X. When this happens, we scrap all of them and flip our regular two cards. Neither of them is white with probability (1 − PW)2. Therefore, the probability of not flipping white cards with Focus X is, again, (1 − PW)2+X, hence the proof.

To simplify the argument, we've considered probabilities with replacements in the previous proof. But it's easy to show that the same result holds, in exact terms, when cast in terms of probabilities without replacements. A brief discussion on the difference between probabilities with and without replacements can be found in the set of notes on Blowpipe.