P = 


 =  = 231;

 =  = 120;

 =  = 73815;


    P =  = 0.3755.      ANSWER
ANSWER

Ikleyn first answer is correct, but her second way is incorrect for it
calculates the probability of choosing them in this order:
1. any woman first from the 22 women, 
2. any woman second from the remaining 21 women,
3. any man third from the 16 men,
4. any man fourth from the remaining 15 men.

But the order in which they are chosen does not matter.

4 people with exactly 2 men also have exactly 2 women.

Choose the 2 men any of "16 CHOOSE 2" or 16C2 = 120 ways
For each of those 120 ways to choose the 2 men, there are 
"22 CHOOSE 2" or 231 ways to choose the 2 women. That
amounts to (120)(231) or 27720 ways.

The number of ways to choose ANY 4 from the 22+16=38 people is
"38 CHOOSE 4" or 38C4 = 73815 ways.

So the probability of choosing 4 with exactly 2 men is
27720 ways out of 73815 or



which reduces to



and which is approximately 37.6% of the time 2 men and 2 women
will be chosen.

Edwin