Probability of other players being dealt a better hand.

Suppose you are playing against 5 other people. If you are dealt 3-of-a-kind, what is the probability that one or more of your 5 opponents was dealt a higher hand than 3-of-a-kind? (Make the simplifying assumption that all players’ hands are independent.)

Quote: movemypeeps9

Suppose you are playing against 5 other people. If you are dealt 3-of-a-kind, what is the probability that one or more of your 5 opponents was dealt a higher hand than 3-of-a-kind? (Make the simplifying assumption that all players’ hands are independent.)

I don't think you can make that assumption. Not sure what game you're playing, but apparently not one with 2 hole cards. However, in any single deck game the effect of removal (of your hand) will be more apparent, but it affects the percentages in multi deck games too.

If you have 3oak, it will be slightly more likely someone else has 3oak as well. They might outrank you. Your 3oak will make straights less likely, though how much less depends on the rank of your 3oak; essential straight cards like 5's and 10's will affect straights the most.. Flushes mostly unaffected, since you hold 1 of 3 suits, plus 2-4 other cards, probably an average of any random hand. Full houses and 4oak slightly more likely. Sf slightly less likely, again depending on the rank of your 3oak.

If we're talking draw poker, it's about 8:1 against your dealt trips not being the best hand (with five opponents). However, the actual number would vary depending on whether you had, say, 222 or AAA.

And what cards have already been dealt definitely matters. As an example, in pai gow poker, if you have two pair in your seven-card hand, that alone increases the chance of a randomly chosen other dealt hand also containing two pair by twenty percent.

Going back to your example, if we're talking a five-card poker hand, even the other two cards matter. For instance, QQQ32 will be beaten more often than QQQ105.