Re: Probabilty Question
From: Matthew Jackson (majst46_at_intergate.com)
Date: 09/17/04
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Date: Fri, 17 Sep 2004 19:36:25 +0000 (UTC)
My interpretation of the question that you have asked is a little
different to that of the others who have posted here, so, on the off
chance that I'm right, I'll make a stab at answering the question I
think you asked:
I think that you asked whether you can tell what the heads probability
of a coin toss is, and whether you can verify that the probability of
a head is 0.5.
The answer to this is "no, but...". To clarify, I will introduce some
terminology. The probability p that a head comes up is called a
paramater. It is generally not possible to observe a parameter
directly. Instead, you can observe the results of your experiment, and
attempt to deduce properties of the parameter from the results.
One approach to doing this is by means of an hypothesis test. You
formulate the null hypothesis (that p=0.5), and the alternative
(something like "p is not equal to 0.5"), and then decide whether the
evidence (1000 coin tosses for example) justifies rejecting the null
hypothesis. Of course statistical anomolies occur all the time, and
you might reject the null hypothesis when in fact p does equal 0.5. It
is the job of the statistician who designs the experiment to ensure
that the likelihood of this happening is low.
Another approach is to construct a confidence interval for the
parameter. This interval is a function of the observed data, and
provides a range of possible values for the parameter. The
interpretation is that you are somewhat/fairly/highly/extremely
confident that the value of the parameter is indeed in the interval
(depending on how you construct the interval). Unless your interval
is trivial, you cannot be totally confident that the value of the
paramater is in the interval.
A different approach is to use the Bayesian technique of assigning a
"prior" distribution on the value of the paramater. This distribution
represents the experimenter's assumptions and beliefs about the value
of the paramater. As more experiments are carried out, the prior
distribution is updated (via Bayes' Theorem) and becomes the
"posterior" distribution.
Generally speaking, the questions that think you asked are related to
the underlying problem of estimation -- how do you estimate the value
of the parameter, and how sure can you be that your estimate is close?
Estimation is a huge area of statistice, and cannot be summarised
here. (Hypothesis testing is not generally considered an area of
estimation theory, as it seeks to answer true/false questions, rather
that output approximations to an unknown parameter).
I hope that this is helpful,
Matthew
On 03 Apr 2001, James Martin wrote:
><pre>
>Can anyone tell me if it is possible to find the probability that a given
>situation will work out in the correct probability? For example, what is
>the probability that a coin toss will actualy yield a 1:1 ratio?
>
>Thanks,
>Rob Martin
>
>
>
></pre>
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