Numinations — May, 2000

What are the Odds?

© 2000, by Gary D. Campbell

   The chances of an outcome depend on the mechanism or system that selects one outcome from all the possible outcomes.  If one numbered ticket stub is selected at random from a hundred, your stub has one chance in a hundred of being drawn, all things being equal.  If your stub is larger, or is folded, or was on top, it might change the “all things being equal” condition, and alter your chances from 1/100 to somewhat more or less than that.  It depends on the actual selection process.

   When chance is involved it means that there are factors or unknowns that we cannot take into account.  Selection processes can be very complex.  When twelve horses run a race, the winner is the horse that first crosses the finish line.  This is a function of the quickness and condition of the horse, the weight and behavior of its jockey, how good a start the horse was able to achieve, the conditions of the track over which it ran as compared to the other horses, and the way the horses interacted with each other during the race.  The combination of these factors is partly predictable and partly luck.  The final outcome can’t be predicted with any certainty.  This is true for the outcome of any complex or chaotic selection process.

   Probability is a funny thing.  An accurate estimate depends upon a set of assumptions being true.  In many real applications of probability, the assumptions aren’t true.  Other times there may be information available that should be used to adjust the assumptions.  The classic assumption in probability theory is that each possible outcome of a trial is just as likely as any other.  When a number of trials are performed, each trial may have a dependence upon the previous trial, or it may be completely independent.  We often encounter only a single trial, but we must make a decision as though the odds were calculated on the basis of many trials.

   Consider a scenario:  A king leads you to a room with three vaults behind closed doors.  He tells you that two of the vaults are empty and one is filled with treasure.  He invites you to pick a door and have whatever is behind it.  After making your choice, but before the door is opened, the king opens one of the other two doors and reveals an empty vault.  He now says that you may have all of the treasure behind the door you have already picked, or you may have 75% of the treasure behind the other closed door.  Which is the better deal?

   The better deal depends on the true odds, given all the information at your disposal.  When you first picked a door, you had one chance in three of picking the treasure.  That means there is a two out of three chance that the treasure was behind one of the other two doors.  The King has just shown you which of those doors you shouldn’t pick, so all you have to do is pick the other one.  The odds have gone from 2/3 that the treasure is behind one of the other two doors to 2/3 that it is behind a single door.  So, you can have 100% of a 1/3 chance (by sticking with your original choice), or 75% of a 2/3 chance (by switching to the other closed door).  Your expected share improves from .33 to .50 if you switch.

   Look at it this way:  There are three doors.  Suppose you initially numbered them at random, 1, 2, 3.  You eliminate number one from consideration.  You open door number two to see what’s there.  If number two has the treasure, you take it.  If not, you take your chances with door number three.  One third of the time you will toss out the treasure with door number one.  One third of the time you get the treasure behind door number two.  And, one third of the time the treasure will be behind door number three.  If it is, you also get it, because there will have been nothing behind door number two, and so you switch to door number three.  This procedure always gives you 2/3 odds of getting the treasure.

   This is the same scenario as with the King, except that the King picks one of the doors and shows you an empty vault (he also charges you 25% for this service!).  Either he knew which of the doors to open, or he was just lucky, but it doesn’t make any difference.  He still resolves the 50/50 uncertainty between which of the other two doors you should pick (the door you first chose was really for the purposes of ruling it out; if the King is willing to oblige, you always intend to switch to one of the other two doors).

   Three men are on death row:  Able, Baker, and Charlie.  The warden announces that one of them is to be executed tomorrow.  Able knows the warden, and asks him which of Baker and Charlie will not be executed.  The warden confides that Baker will not be executed.  What are Able’s chances now?  (Unchanged, with a 1/3 chance of being executed.)  What are Charlie’s chances?  (Changed from a 1/3 chance to a 2/3 chance of being executed, since Baker’s chances changed to zero.)

   Three astronauts, Able, Baker, and Charlie are told that two of their group have been chosen to go on the next mission.  Able figures that his chances for the mission are two out of three.  Now, suppose Able gets his supervisor to confide that Baker is going for sure, does this affect the odds that Able is going?  (No.)  What are the odds that Charlie is going? (Changed from a 2/3 chance to a 1/3 chance.)

   For some, these statements may be counter intuitive.  They are certainly worth Numinating about, because real life often offers us choices or phenomena that can be decided or explained only when we understand that they are probabilistic, and what the odds actually are.  When a statistical model is the very best way we have to explain the behavior of something, it simply means we are dealing with uncertainties that follow a pattern.  We can improve the situation by resolving some of the uncertainties, or we can make the situation worse by applying the pattern wrongly, or by applying the wrong pattern.

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