just wanted to make sure im thinking about this correctly

this is for use in plo5

q1. board is JT2, i want to know odds of ANY straight completing by river, this is some kind of permutation, yes? since turn brings in more draws.
anyway , i brute forced it with odds oracle.it seems generally its about 80%

we know our odds of a set boating up is ~30%

q2. what is the probability of getting to the river with our set EITHER boating up, or fading all straights?

which function is this on the calculator, again?

also, out of curiosity, how would we do q1 longhand?

TY !

Q1: a straight becomes possible if at least one {7,8,9,Q,K,A} comes OR if two ranks of {3,4,5,6} come.

Of the 49 remaining cards, 25 aren't in that first group, so from the C(49,2) combos we can subtract C(25,2). To include the backdoor straights we add back C(4,2)*4² because we're choosing 2 ranks from those four and then their suits.

All told, P(straight possibility) = 1 - [C(25,2) - C(4,2)4²]/C(49,2) = 81/98 ≈ .8265


Q2: Assuming we only know two of Hero's hole cards (ie the pair forming a set), we're operating with 47 unknown cards. Suppose Hero has 22.

The boat/quad possibilities are any J/T/2 or running xx. The probability is 1-[C(40,2) - 10*C(4,2)]/C(47,2) = 361/1081 ≈ .334

On this board, there's actually no way to fade all straights while also missing a boat or quads. So the chance of improving or fading all straights is the same as the chance of improving: 33.4%.