Re: Math Midget seeks help with odds

chrystalia wrote:

Greetings:

I am not a math professional, but I am hoping someone will help me. My
partner and I are running what could be termed "binary prediction
tests", in that we are testing subjects on their ability to predict the
outcome of coin tosses (heads or tails) and the colors of unseen
playing cards (red or black). The tests are, obviously, designed to
determine the presence or absence of so-called psychic abilities
(precognition, mainly). We have data sets that obviously show non
random affects. The question is, how unlikely are the divergences? For
example, we have one subject who has tossed a total of 222 tails
(longest streak 23 consecutive tails) and 135 heads in a single series
of tosses, and consistently tosses a head/tail ratio of 2:1.

If someone "consistently" tosses a head/tail ratio of 2:1 then either
the coin is biased, your procedure is somehow otherwise flawed, or the
subject is tricking you.

From the binomial probability distribution (see e.g.
http://mathworld.wolfram.com/BinomialDistribution.html) the probability
of getting exactly T tails in N tosses of a fair coin is

N!/(T! * (N-T)!) * (1/2)^N

(and similarly for heads).

However, as N gets larger the probability of tossing ANY exact given
number of tails, even if close to N/2, gets smaller and smaller, so
this is not necessarily a good measure of "unlikeliness". Instead you
could calculate the probability of getting *at least* the given number
of tails, which you can do by summing the above over the appropriate
range of T.

The probability of getting at least 222 tails in 357 tosses works out
as about 1 in 416,000. This is small enough to raise questions about
the validity of the procedure you are using. But remember also that the
more trials you run the more likely it is that at least one will
exhibit this kind of unlikeliness. If the probability of success is p
(in this example p = 1/416,000) then the probability of at least one
success in X trials is 1 - (1 - p)^X. So, if you ran 1,000 such trials
you would expect at least one such outcome with probability 1/416.



We have another subject who made 213 correct guesses and 151
incorrect guesses of playing card colors for a total of 364 guesses. We
would like to be able to analyse binary trials of this type. Are there
simple formulas we can use to do so?

Thank You for any assistance you may have to offer :-)
Chrystalia

.

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