Math Help - Probability

1. Probability

I'm not sure how to solve this problem...so here goes:

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 the 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 probabilty that it was that of a male?

Thanx for the help!

2. Hello,
Originally Posted by dolphinlover
I'm not sure how to solve this problem...so here goes:

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 the 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 probabilty that it was that of a male?

Thanx for the help!
Let M be the event for "it is a male"
Let F be the event for "it is a female"
Let P be the event for "it is positive"
Let N be the event for "it is negative"

Note that M is non F, and P is non N.

You're looking for P(M/N), that is to say "the probability that it is a male given that it is negative.
You know that P(M)=5/20=0.25, since there are 5 male out of 20 people.
And P(F)=15/20=0.75.
You know that P(P/M)=0.4 since 40% of the males react positively (it's equivalent to saying that there is a probability of 40% that a man reacts positively).
Since $P=\bar N$, we can say that $P(N/M)=1-P(P/M)=0.6$
We know that P(P/F)=0.7 and with the same reasoning, $P(N/F)=1-P(P/F)=0.3$

These is all the information given by the text.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now use the formula for conditional probability :
$P(M/N)=\frac{P(M \cap N)}{P(N)}$
But we also know that $P(N/M)=\frac{P(M \cap N)}{P(M)} \implies P(M \cap N)=P(N/M)P(M)$

Hence $\boxed{P(M/N)=\frac{P(N/M)P(M)}{P(N)}}$

The law of total probability gives us this formula :
$P(N)=P(N/M)P(M)+P(N/F)P(F)=\dots=0.525$

You can now calculate $P(M/N)$

3. Originally Posted by dolphinlover
I'm not sure how to solve this problem...so here goes:

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 the 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 probabilty that it was that of a male?

Thanx for the help!
A tree diagram is the simplest approach here I think.

The first two branches are M (probability = 1/4) and F (probability = 3/4). The branches coming from these are the positive and negative branches.

I get $\Pr(M | -ve) = \frac{\frac{1}{4} \cdot \frac{6}{10}}{\frac{1}{4} \cdot \frac{6}{10} + \frac{3}{4} \cdot \frac{3}{10}} = \frac{2}{5}$.