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### Probability of Independent Events

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Date: 02/07/97 at 18:36:51
From: Carl J. LaGrassa
Subject: Probability question

Given that the probability of an event is known, is there a formula
which can give the probability of an event in a given number of
trials?

To use specific examples, I am interested in the probability of an
event occurring in 16k, 32k, and 64k when the probability of
occurrence is 1/32000.

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Date: 02/08/97 at 05:31:03
From: Doctor Mitteldorf
Subject: Re: Probability question

Dear Carl,

There aren't many formulas for probability questions.  The big
question here is whether your 16k copies of this problem are all
independent of one another.  Does the occurrence of the event in one
place forebode an enhanced probability that it will appear in another
place?

With that caveat, I can give you a formula: for independent events,
each with probability p, the probability of n such events is:

1 - (1-p)^n

For your first example, this is 1-(31999/32000)^16000.  Think of it
this way: The first one has a (31999/32000) chance of not failing. If
it doesn't fail, the second one has a (31999/32000) chance, etc. If p
is small, this turns out to be approximately equal to e to the power
(-n/p).

- Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

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Date: 02/08/97 at 22:02:25
From: Carl J. LaGrassa
Subject: Re: Probability question

I'm afraid I don't understand the answer.  I was speaking of
independent events.  I do not understand the notation "^" or the
reference to "e".

Specifically, in video poker a royal flush will occur randomly with a
frequency of 1/32000 hands played.  I am looking for the probability
of getting an occurrence in a given sample size.
```

```
Date: 02/09/97 at 06:21:46
From: Doctor Mitteldorf
Subject: Re: Probability question

Dear Carl,

The symbol ^ is used on computers to indicate "to the power of".
x^2 means x squared.  I use it here because it's not easy to send
raised exponents over the internet.

The point is simply that if two events are independent, you can
multiply their separate probabilities to get the probability of both
happening.  If the probability of a single hand coming out not-a-
royal-flush is 31999/32000, then the probability of two coming out
not-a-royal-flush is (31999/32000)^2 and the probability of three
coming out not-a-royal-flush is (31999/32000)^3.

To get the probability that none of the 16000 trials will come out a
royal flush, simply multiply sixteen thousand times the probability
that each separately will come out not-a-royal-flush.  This can be
written (31999/32000)^16000.  Take the number (31999/32000) and
multiply it by itself 16000 times.

The statement that there will be at least one royal flush in the 16000
trials is just the opposite of the statement "none of the trials will
produce a royal flush."  So you can get that probability by
subtracting that product from 1.

--------------------------------------------

e is the base of natural logarithms and is approximately equal to
2.71828182845... One definition of e is that if you take larger and
larger numbers n and calculate the nth power of (1+1/n), the answers
come out closer and closer to e.  This is something you can try on a
calculator.

It also works that the nth power of (1-1/n) comes closer and closer to
1/e as n gets larger and larger.  Try it!

You can use this last fact to come up with an approximate expression
for (31999/32000)^(16000).  The thing you're raising to a power,
(31999/32000), is very close to 1.  Think of it as 1-1/n, where
n = 32000.  You're not raising it to the nth power, but to half n.
We use the fact that anything to the power of half n is just the
square root of the thing raised to the nth power.  (Try proving that
by multiplying two copies of the thing together, and simplifying what
you get.)

So (31999/32000)^16000 is very close to e^-0.5, or 1/e raised to the
power 1/2.

Thanks for writing back.  Please let me know if this is clear now.

-Doctor Mitteldorf,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

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