When playing dice the probability p of rolling a 1 in a single roll
is 1/6, the probability q of not rolling a 1 is q = 1-p = 5/6. The
probability of rolling two 1's in two rolls is p^2 = 1/36, because
the two rolls are independent events. More interesting is the
probability of not rolling any 1's at all. Call this success.
Similarly, for two rolls, this probability is q^2, or 5/36, because
the rolling events are independent. It does not matter the order in
which order the die are rolled or whether they are rolled
simultaneously. For n casts of the die, the probability of success is
q^n. The probability of failure is thus 1-q^n = 1-(1-p)^n.
Solo Russian Roulette can involve spinning a cylinder of a pistol
with a single round in it, the player aiming the gun at his own head,
and pulling the trigger. Call this an event. In the case of a six
shooter, the probability p of the player killing himself, failure, in
one event is 1/6. The probability of success in a one event game is
then q = 1-p = 5/6.
If a cylinder spin occurs in each event, then the probabilities of
success or failure in any two events are independent of each other.
The probability of success when playing a two event game, given both
events are independent, is as in dice, q^2 = 25/36. The probability
of success in an n event game is q^n. The probability of failure is
thus 1-q^n. If p is the probability of failure in a single event,
then 1-(1-p)^n is the probability of failure in an n event game. The
only difference between this form of Russian roulette and dice is the
events of a game must stop upon the first failure event. The
aggregate events can be looked upon as dependent in that sense. The
events always occur one at a time and the latter events do not occur
once a failure is encountered. However, the same can be true of the
dice game. If the dice are rolled one at a time, then once a 1 is
encountered it is not necessary to continue rolling the dice because
the outcome is then already determined.
The formula 1-q^n provides some not common understanding of games of
chance, risk taking, and product liability. This understanding comes
from the fact that in the limit, as n approaches infinity, for *any*
positive q less than 1, q^n rapidly approaches zero. This is
expressed as:
lim n->inf q^n = 0 for all q<=0<1
In other words, if there is any possibility of failure, i.e. q<1,
then repeated events eventually, much more quickly than ordinary
common sense dictates, result in failure. For example, if the
probability of a catastrophe when drinking and driving once is 1/1000
= .001, then the probability of no catastrophe in 100 such events is
1-.999^100 = 0.095, or about 1%. Each event is independent, yet the
combined effect of repeating events has a dependent nature. This is
sometimes called the Russian roulette effect.
Similarly, if a condom brand has a 1% chance of failure, then 100
uses results in a probability of failure at some time in those uses
of 1-.99^100 = 0.634 = 63%.
If every critical component of a rocket has a very small chance of
failure, say 1/10000, but there are 1000 such components, then the
probability of system failure is 1-(0.9999)^1000 = 0.095, or about 10%.
If a product, like a vehicle, is used N=50 minutes a day by
M=10,000,000 people, and the probability of failure in any given
minute is p, then the probability of some failure P in a year is
given by:
P = 1-(1-p)^N*M*365 = 1-q^182500000000
Is is easy to see then, that for a product used by many people that,
as time goes on, the number of opportunities for failure, n=(years)
*N*M*365, becomes very large. No matter how close q is to 1, q^n then
approaches zero. If anything can go wrong it will. This is Murphy's
law.
Similarly, each of the bets of a gambler is an independent event.
However, all gamblers have a limited purse, even when credit is
available. When a purse runs out then the gambler is done. If a
gambler plays against less than favorable odds, no matter how small
the margin, he eventually loses his purse, all he has. This makes the
independent events dependent in that context, even though a single
failure is not a total failure. This happens much more quickly than
often understood, even with typical house odds, and in a
psychologically unfortunate manner. This is described in detail here:
http://mtaonline.net/~hheffner/Gambling.pdf
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
