What are the Odds......?

            A friend of mine, who is in the gaming industry, posted the following on Facebook this past week:

            What are the odds? Six pairs of dress socks in the dryer. I didn't get a match until the fifth sock I pulled out.

            You simply can't be in the gaming industry, have a lot of friends in the gaming industry and expect anything but a serious math discussion to follow.  One of his friends, also in the gaming industry, beat me to the response and posted that the odds were about 77 to 1 and suggested a payback of 70 to 1.  Unfortunately, in his haste, he made a few errors in his calculation.  This didn't smell right to me as it just didn't seem that outrageous for this to occur.

            So, I did what comes so naturally to me.  I opened an MS Excel spreadsheet and began working on a math solution.  To verify the number I got, I quickly wrote a program (okay, I was obviously bored on this particular evening with all those Hallmark X-mas specials on cable!).  Both methods provided the same answer, so I figured I was safe to post it.  In short, the probability is 24.24%.  Anyone who wants to offer up a 70 to 1 payback is going to lose a lot of money.

            The calculation for this problem goes as follows.  We have a total of 12 socks in six pairs.  The first four socks came from different pairs and the 5th sock was the match to one of the other four.  So, the first sock can be any one of the 12 (probability = 1.0).  The second sock can be any of the socks EXCEPT the match to the first one.  So, 10 out of the 11 socks will fit this criteria (probability = 0.9091).  The third sock, can be any sock EXCEPT either of the two that match the first ones.  So, there are 8 out of 10 socks remaining that it can be (probability = 0.80).  The fourth sock can be any of 6 of the remaining 9 (probability = 0.6667).  Finally, the 5th sock MUST be one of the 4 that match any of the first 4.  So, 4 out of 8 meet this definition (probability of 0.50).  We multiply these five probabilities together to arrive at 24.24%

            What does this have to do with anything in the casino?  It is just a relatively real-life example of how people do not do such a good job recognizing the true odds of things.  The original post would seem to indicate that the write was under the impression that what occurred was rather rare.  So much so, that another knowledgeable person doesn't find it odd to state that the probability is between 1% and 2%.  This seemed reasonable to both of these people.   Would the original poster put that up if they had known that what had just occurred was hardly an uncommon event.  Nobody gets all that surprised when dealt a High Pair in video poker and the probability of that is just under 15%.

            As you read the first couple of paragraphs of this column, what did you think the probability of the event occurring was?  When you read that someone stated it was 77 to 1, did this seem reasonable?  Does my correct response of a probability of 24+% surprise you?  Based on the information I provided, many of you may be surprised to realize that once the first four socks came from different pairs, that the probability that the 5th sock would match one of the first four is 50%.  This also means that the probability of the 5th sock coming from one of the two remaining pairs is ALSO 50%.  

            As surprising as a result my friend found this to be, I guess he would find it equally surprising that the outcome he got was actually the most common outcome!  Both having the 4th sock and the 5th sock be the first that matches one of the prior ones will occur 24.24% of the time.  So, this extraordinary event which was so surprising that it warranted being posted on Facebook really turned out to be not just routine, but in fact the likely outcome.   This is frequently the case, even when you hear someone in the casino sound quite certain that the deck must be loaded and/or the casino must be cheating because something they perceive as being so rare seems to be occurring in relative abundance.

            I'm sure my friend wound up with a far different discussion than he anticipated.  In the end, my friend decided to call the game "Ultimate Texas Fluff and Fold'em".  Since I have a (bad) habit of just shoving all my socks in the sock draw (unmatched), I suggested "No Fold'em".  In the end the whole conversation was summed up by a 3rd party who mused "For most people, starting a sentence off with 'What are the odds' would be just a figure of speech.  Not for us gaming analysts.......