Poker coincidences
In the play Rosencrantz and Guildenstern are dead by Tom Stoppard, Rosencrantz and Guildenstern flip a coin repeatedly for money:
Rosencrantz: Heads.
(He picks it up and puts it in his money bag. The process is repeated.)
Heads. (Again.)
Rosencrantz: Heads. (Again.)
Heads. (Again.)
Guildenstern (flipping a coin): There is an art to the building up of suspense.
Rosencrantz: Heads.
Guildenstern (flipping another): Though it can be done by luck alone.
Rosencrantz: Heads.
Guildenstern: If that’s the word I’m after.
Rosencrantz (raises his head at Guildenstern): Seventy-six love.
This isn’t of enormous relevance to this article, but I find it difficult to work into any conversation that I played Rosencrantz in my school production of Hamlet, just a few decades ago . .
I was however playing a game of social poker recently and one of the players asked me whether I knew the probability of picking up the same hand twice in a row. Yes, Jack, as a matter of fact I do. Well ok, only to the nearest order of magnitude, just off the top of my hand, in real time at the table. The exact odds are 1 in 1326. Not quite as unlikely as flipping heads 76 times in a row, as in R & G are dead, but still very improbable.
The end
But wait, dear reader. If you have a few more minutes to read on, I do have a bit more to say on this topic. Picking up the same hand twice (or more) in a row is just a coincidence, incidental to the game of poker itself. And it only really ‘counts’ if I had turned to Jack and offered him 1000 to 1 odds that his next two hands wouldn’t be the same. Then if Jack picks up, say, 7 of spades & 3 of diamonds, twice in a row, that would be saying something, and Jack would win a big wager (but lose in the long run, given I’d offered Jack less than 1326 to 1).
To give an analogy, suppose your first name is Muhammad. You decide to take the bus into the city, and you start chatting to a stranger on the bus whose name also turns out to be Muhammad. What a coincidence! Well, not really, because the ‘sample space’ (as us statisticians say) of possible coincidences is huge. The stranger could have been wearing the same shoes, have gone to the same university, or some other unrelated coincidence could occur later that day in the city. And besides, ‘Muhammad’ is the most common first name in the World!
Or suppose your family name is ‘Wang’. You’re on the bus into the city and strike up a conversation with a stranger (you may already see where this is going ..) who also happens to be called Mr Wang. Similarly, big deal, ‘Wang’ is the most common family name in the World . .
However, if your name is Muhammad Wang, and you bump into a stranger who’s perhaps the only other guy in the state also called Muhammad Wang, ok that’s a notable coincidence! But only if you called it out before you left home . .
Anyway, I digress, back to the slightly smaller statistical sample space of poker coincidences. As noted above, the probability of a particular player’s next hand being the same as the previous one is 1 in 1326. We could estimate this in real time at the table, depending on the number of whiskeys imbibed of course, because there are 52 cards in the deck, which is approximately 50. So the probability of the first card being dealt matching one from the previous hand is 2 in 50 and of the second card then also being a match is 1 in 50. Multiplying these together we get 2 in 2500 or 1 in 1250. Close enough to 1 in 1326.
However. Poker players will often call 44 (a diamond and a club) and 44 (a heart and a spade) the ‘same’ hand. These two hands are functionally the same in poker, preflop at least. Let’s call this a non-exact match. This brings down the odds considerably. There are 4 combinations of every suited hand, 6 of every pocket pair and 12 of every off-suit hand, reducing the odds of a non-exact match to 1 in 332, 1 in 221 and 1 in 111 respectively. The weighted average is 1 in 136. This is actually a bit difficult to estimate in real time, though we could note that ignoring suits there are 13 * 13 = 169 possible starting hands. 1 in 169 is not dissimilar to 1 in 136.
The next point to note is that we would be equally surprised if any one of the other 7 players around the table that night, not necessarily Jack, picked up the same hand twice in a row. This shortens the odds of someone hitting the same (exact) hand twice in a row by almost exactly a factor of 8. So odds of 1 in 1326 becomes odds of 1 in 166.
The reason for the ‘almost’ is that once in a blue moon two (or more) players will get a repeat hand. The true odds of at least one player getting a repeat hand is 1 in 167. If we were estimating this in real time we might divide our earlier guesstimate of 1 in 1250 by 8 and get 1 in 150, give or take.
The odds against at least one of the 8 players picking up a non-exact match is 1 in 17 (again, dividing our earlier estimate of 1 in 136 by a factor of 8). We’re now getting to into the ‘hardly surprising’ territory!
Finally, let’s consider that in the course of a night we might play 100 hands of poker. If Jack gets the same hand on, say, deal #56 and deal #57 that might still feel ‘surprising’. But not really. If the odds are 1 in 1326 on one comparison, then you can divide the odds by 100 and say the odds are about 1 in 13 given 99 consecutive comparisons. (Exact probability 1-(1–0.075%)⁹⁹ or 1 in 14). A coincidence, but not a huge one.
Dividing the odds by 100 becomes a bit inaccurate in the case of 8 players (viz. arguing that if for 2 hands the odds are 1 in 167 then for 100 hands the odds must be 1 in 1.67). This would suggest that after 168 hands we’d be certain to have a match, but we’re double counting multiple matches. The true probability is closer to 50% that at least one of the 8 players has an exact ‘repeat’ in the course of the evening.
If all these statistics have left your head spinning, the sole point I’m trying to make is that an observation that started out seeming implausible turns out to be commonplace. It’s also important to define the coincidence accurately: do we mean just Jack, or any player, and do we mean just this hand or any hand tonight?
The same applies to other patterns we might see at the poker table or elsewhere in life. Maybe we notice that we’re only picking up red cards (diamonds or hearts) or only cards that are prime numbers, or whatever; the possible coincidences are endless. But it’s only truly noteworthy if we anticipated these patterns a priori.
A final statistic is that in the course of 100 deals, the 8 players will play 800 hands in total but only about 600 unique hands (out of the 1326 that are possible). This means that 25% of the hands that we play will be picked up by someone else that night. The question is: will we play that hand more skilfully?