Males and females are observed to react differently to a given set of
circumstances. It has been observed that 70% of the females react
positively to these circumstances, whereas only 40% of males react
positively. A group of 20 people, 15 female and 5 male, was subjected
to these circumstances, and the subjects were asked to describe their
reactions on a written questionnaire. A response picked at random from
the 20 was negative. What is the probability that it was of a male?

Thanks in advance!

roadapples,

Following the trail of information we are given, we see that out of 15
women, 70% of them should give a positive response.  This would give
us 10.5 women with a positive result, and 4.5 women with a negative
result (I'll ignore the question of what 0.5 women is supposed to be,
since it is ultimately not relevant to the final answer)  Meanwhile,
out of 5 men, 40% would have a positive reaction, meaning that 2 men
would give a positive reaction, while 3 would give a negative one. 
This means we have 7.5 total negative reactions, 3 of which come from
males.  Dividing 3 by 7.5 gives us the result that 40% of all of the
negative responses in this study should come from males.  So the
probability that a randomly selected negative response from a pool of
15 females and 5 men is from a male is 40%.

Thanks for the answer!

I agree with hailstorm-ga's answer.  The mathematical name for this
sort of argument is Bayesian inference.  There is a specific formula
(Bayes formula) that expresses the conditional probability.

To apply Bayes formula (as hailstrom-ga did above in words), let:

M = response from male

F = response from female (complementary event to above)

N = response is negative

Then the application of Bayes formula:

P( M | N ) = 

           P( N | M ) * P( M )
-------------------------------------------
   P( N | M )*P( M ) + P( N | F )*P( F )

where the denominator represents an expansion of P( N ).

Plugging in the various data given in the problem, Bayes formula
evaluates to:

              (.60)*(.25)
-------------------------------------------
      (.60)*(.25) + (.30)*(.75)

= (.15)/(.375) = 40%

Bayes formula can be derived easily from the definition of conditional
probability.

regards, mathtalk-ga

Wow, mathtalk-ga,
you surely likes math!

Me too, although, I'm not very good at math anymore :(
I'm going to sue my teachers in high school, I wish you're my teacher, LOL!