Hi,
On 2-10-2011 14:25, Horace Heffner wrote:
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.
This should of course be read as:
Similarly, for two rolls, this probability is q^2, or *25/36*,
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%.
This should of course be read as:
then the probability of *a* catastrophe in 100 such events is 1-.999^100
= 0.095, or about 1%.
As earlier said: The probability of failure is 1-q^n = 1-(1-p)^n.
Each event is independent, yet the combined effect of repeating events
has a dependent nature. This is sometimes called the Russian roulette
effect.
No, this suggests that the pistol has a memory, which is of course false.
The Russian roulette effect is a mental process that happens between the
ears of the person (with his a memory!) spinning the cylinder.
This brings back memories during my student time when we had a similar
discussion among students about throwing a dice so we decided to perform
an experiment and it turned out that statistically after throwing a
thousand times the dice this resulted in an almost equal number of times
that all the numbers of the dice were thrown.
That's also the reason why well performed surveys always require a
representative amount of people to be interviewed.
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%.
Correct, but as you know condoms are for obvious reasons meant for
single use only ;-)
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.
True, and that's also the reason why only MTBF (Mean Time Between
Failure) is used for critical parts, systems, etc. for military
vehicles, aircraft and spacecraft.
So a MTBF for a vehicle of once in 1 year could statistically result
with P = 1-q^n in 1-(1-(1/MTBF))^M = 1-(1-(3600*24*365))^10,000,000 =
0,2717 or 27,17 % total failure once a year.
Kind regards,
MoB
