i’ve seen this question come up before in deckbuilding philosophy in rs-pk. that is, which is better: poke ball or master ball? henceforth PB and MB respectively, for simplicity’s sake. in order to best determine the probabilities for these, we need to look at sampling without replacement.
the primary decks that the math is “easy” for are turn 2 stage 1 ex decks, the primary two being scizor ex and turbo manectric ex. for early game, assume a 46 card deck, with 1 basic pokemon in play (e.g. a scyther or an electrike). the odds MB grabs a pokemon would be:
(n choose x)*(T – n choose r – x) / (T choose r)
where n is the number of cards being checked (7 in the case of MB), x is the number of pokemon you are finding the probability of finding (1-3 for basics, 1-4 for evos, and higher numbers for “extraneous” cards), T is the number of cards in the deck before using MB (e.g. at the start, this would be 46 – kind of, more on that later), and r is the number of cards that can be what we’re looking for (e.g. for basics it would be 3, and for evos it would be 4). so, for a deck like turbo scizor ex, assume we play 4-4 scizor and no other pokemon, and open a scyther with no other basics. we have no scizor ex in hand, but we have either a MB or a PB, and the rest of the cards are irrelevant (assume, say, 5 energies).
now, for decks playing these t2 stage 1 pokemon, the idea is usually that you aim to get a stage 1 out, and it’s this stage 1 success rate number we care about in particular. manectric ex decks might not mind grabbing an electrike or a lone mew ex off of a MB, but scizor ex decks in particular would prefer not to, as scyther itself easily grabs 2 more from the deck. we should assume, then, with the manectric ex deck, that its r is equal to 4 and its N is equal to 45 (assume recharge from electrike), and for the scizor ex deck, that its r is equal to 4 and its N is equal to 44, as it assumes using one twin play at the least.
then, for scizor, we can simplify this a bit to:
(7 choose x)*(44 – 7 choose 4 – x) / (44 choose 4)
or, even simpler:
(7 choose x) * (37 choose 4 – x) / (44 choose 4)
to calculate the probability of at least one scizor ex being found off of this initial MB, we then substitute x = 1, 2, 3, and 4, and find the sum of these probabilities:
for x = 1: 0.4007
for x = 2: 0.1030
for x = 3: 0.0095
for x = 4: 0.0003
sum: around 51.35%.
this is very slightly better than PB odds, which are a flat 50%. we see similar numbers for the manectric ex deck:
(7 choose x) * (38 choose 4 – x) / (45 choose 4)
for x = 1: 0.3963
for x = 2: 0.0991
for x = 3: 0.0089
for x = 4: 0.0002
sum: around 50.45%.
again, slightly better than PB odds. MB odds generally increase the deeper one digs into the deck without a hit (for example: imagine playing mary’s request before going for MB in either of these situations, and the number approaches the 55% territory). note also that there is at least a chance that MB grabs a basic, which is not a great consolation but also not nothing, especially in the manectric decks (or on something like arcanine ex / scizor ex with a 1-1 houndoom or a lunarock line).
however, i’ve been glossing over something thus far, which is that this assumes that you aren’t prizing any of your evolution lines. if we increase our N to be around 51 (which assumes that we don’t know beforehand if we’ve prized any cards), we get odds of around 45.68% of hitting the scizor ex. similarly, if we know we’ve prized a scizor ex (so 3 successes and 45 cards in deck), our odds go down to around 40.55%. you can check all of these odds quickly and easily on stat trek.
so, which methodology is correct? i would say that the best is the latter, as you are evaluating the odds when you have no prior knowledge of your prize contents, and obviously once you do know your prize contents then the odds change. so if our true odds for MB hitting a scizor ex from the top 7 cards of our deck totally blind are indeed around 45-46%, then is it just worse than PB?
in scizor ex specifically, it might be. the name of that deck is getting scizor ex out fast and early, disrupting the opponent with trainer cards and stadiums, and refreshing the hand. for the early game, you would need to be several cards deep with no scizor ex in order for MB to have better functional odds than PB. for this reason, i believe that decks that play very light (<10) pokemon counts, PB is the superior option to MB.
however, there are other decks that have to make these considerations. for example, jason klaczynski’s scrambled eggs list plays 15 pokemon (4-4 exeggutor & 3-4 electrode ex). while his odds of hitting, say, an exeggutor off of a MB are the same as the scizor ex’s player, he has more reasonable odds of 66.90% to hit at least one of exeggutor or voltorb off of a MB with 51 unknown cards – likely a fair and welcome tradeoff to the PB. it’s for this reason that i believe that, in decks that play higher pokemon counts but still have tight, stage 1 attacker builds, master ball is often worth the decreased odds for the “best” outcome off the bat. anecdotally, jason said as much when discussing the issue in scrambled eggs specifically.
additional notes:
-PB does allows you to check your deck in its entirety to see your prizes while MB on its own does not – assuming you hit heads of course. this is a minor benefit for PB.
-depending on prizes and composition of the deck, PB allows you hit cards that are “hard” to hit with MB sometimes (if you prize 2 scizor ex, for example, your MB odds become terrible). of course, the flipside is that if this doesn’t happen, that MB is almost always more live (for example: your odds of hitting one of two scizor ex in a 20 card deck is around 59%).
-MB may have uses in other decks – i’ve been toying around with it in fossilbox lately to mixed results. the idea is that running a deck where you’re so likely to at least hit something off of MB is usually good if the cards you’re hitting always have immediate use. for fossilbox, that deck likes using its supporters to do other things than search pokemon, plays a ton of pokemon in the deck, and each one is usually useful (you can, at the very least, evolve something on the bench, or throw it down with strange cave). in pokemon heavy decks like LBS, though, there are many extraneous cards that have no practical use at the moment even if you do hit them off of MB, so despite the high count, it is often not worth using MB as a searcher in these.
-i cannot see a reason why one would play PB in decks outside of dedicated, T2 stage 1 attacker decks.
core enforcer 