You may be wondering how the power calculator computes probabilities. In order to understand the math, we need to discuss binomial coefficients and hypergeometric distributions.

A binomial coeffient (sometimes referred to as "`n` choose `k`") is a way of computing the number of ways to select a subset of items from a larger collection. In the context of a card game, we can use it to compute the number of different `n` card hands drawn from a `N` card deck.

` ((N),(n)) = (N!)/(n!(N-n)!) `

`N =` Cards in deck

`n =` Number of cards drawn

While `((N),(n))` will give us the number of different hands possible to draw, it doesn't directly give us any probabilities. However, we can get the probability of drawing exactly `k` power cards from a deck containing `K` power cards (of `N` cards total) by dividing the number of possible hands containing `k` power cards by the number of total possible hands. This is referred to as a hypergeometric distribution.

` P(k) = (((K),(k))((N-K),(n-k))) / (((N),(n))) `

`K =` Power cards in deck

`k =` Power cards drawn

`N =` Total cards in deck

`n =` Total cards drawn

So, that's useful. `P(k)` will give us the probability
of drawing a hand of exactly `k` power cards. However,
we usually care about the probability of drawing
*at least* `k` power cards, rather than the
probability of drawing exactly `k` power cards. We
can find the probability of drawing at least
`k` power by adding up the probability values for
every value of `k`, up to the maximum number of power
cards.

`P(k >= T) = sum_(i=T)^(K) P(i)`

`K = ` Power cards in deck

`k = ` Power cards drawn

`T = ` Target power cards

`n = ` Total cards drawn

Okay, so now we know how to compute the probability of drawing enough power. But what if we care about multiple kinds of influence? As it turns out, we can generalize the hypergeometric distribution to compute probabilities when we care about multiple different card types drawn. For example, fire influence drawn and time influence drawn.

` P(k_F,k_T) = (((K_F),(k_F)) ((K_T),(k_T)) ((N-K_F-K_T), (n-k_F-k_T))) / (((N),(n))) `

`K_F =` Fire influence cards in deck

`K_T =` Time influence cards in deck

`k_F =` Fire influence cards drawn

`k_T =` Time influence cards drawn

`N =` Total cards in deck

`n =` Total cards drawn

Again, this will give us the probability
`P(k_F, k_T)` of an exact number of
fire and time, but we usually care about drawing
*at least* that much, so we'll need to
add together all the possible combinations of at
least those target numbers.

` P(k_F>=T_F, k_T>=T_T) = sum_(i=T_F)^(K_F) sum_(j=T_T)^(K_T) P(i, j) `

Now we're getting somewhere! However, the above math
is still a simplified version of what the power
calculator *actually* does. All the above
math assumes that individual cards provide one kind
of influence, but not more than one kind of influence,
or both power and influence.
As we know, in reality, cards can provide multiple
kinds of things:
perhaps power *and* time influence,
or power *and*
time influence *and* fire influence.

In order to account for this, the Power Calculator first groups cards according to the sorts of resources they produce. After generating these groupings, it will enumerate all possible combinations of those grouping which — when combined — will generate enough power and influence to satisfy the target power and influence cost. For each of those enumerated combinations, it will compute the probability of drawing that specific number of sources from each group. Finally, it will sum those probability to find an overall probability of satisfying the target cost.

As an example, consider the case
where we have a deck with three groups of
power/influence sources: time sigils (`1T`),
fire sigils (`1F`), and additional power sources
which provide both fire and time (`1FT`).
Now, say we want to find the probability of being able
to play a card costing `2FT`. We can sum up the
possible combination of grouped cards which will provide
enough power and influence to satisfy the overall cost:

` P_(2FT) = {: (P(k_(1FT) >= 2, k_(1T) >= 0, k_(1F) >= 0)), (+), (P(k_(1FT) = 1, k_(1T) >= 1, k_(1F) >= 0)), (+), (P(k_(1FT) = 1, k_(1T) = 0, k_(1F) >= 1)), (+), (P(k_(1FT) = 0, k_(1T) >= 1, k_(1F) >= 1)) :} `

Thus, the calculator will compute
the probability for each specific combination
of grouped cards which happens to satisfy the
target cost, and then sum those probabilities
into an the overall probability value.
**Eternal**.
If you choose to re-draw, your starting hand is guaranteed to include
either 2, 3 or 4 power cards (with an equal chance of each).

This raises your chances of having more **Influence** to
start with - so (generally speaking) your odds may be ever-so-slightly
higher than what is actually depicted on the charts in the **Eternal
Power Calculator**.
**Power **cost
associated with them. * Seek Power *and the
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