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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