Probability that first and last are same gender? There are twelve members on a team, $8$ women and $4$ men. Out of these $12$, a $4$-person relay team is being chosen. What is the probability that the first and last leg of the race are the same gender?
I started working on this, I though it might be a combination but I cant seem to figure it out.
 A: Approach via multiplication rule of probability and breaking into cases.
"First and last leg of race are the same gender" is equivalent to "First and last are both men or first and last are both women."
Let $M_f, M_l, W_f, W_l$ represent the events man first, man last, woman first, woman last respectively.
The probability you are looking for them is:  $Pr((M_f\cap M_l)\cup(W_f\cap W_l)) = Pr(M_f\cap M_l)+Pr(W_f\cap W_l)~~~~~(\star)$

Multiplication principle of probability:
  $$Pr(A\cap B) = Pr(A)\cdot Pr(B|A)$$

Continuing from $(\star)$ we have $= Pr(M_f)Pr(M_l|M_f)+Pr(W_f)Pr(W_l|W_f)$
Fill in the missing information above and complete the arithmetic to find the final answer.

 $Pr(M_f)=\frac{4}{12}$ since when picking the first position, four of the twelve people available for it are men.  $Pr(M_l|M_f) = \frac{3}{11}$ since we can choose the last person next after having picked the first.  If the first person was a male, then there are three men left who can be in the last spot out of the eleven available remaining people.


Alternatively, approach directly via counting.  Let our sample space be all ways to organize the four people on the team where order is important.  Here we will have $|S|=\frac{12!}{8!}=P(12,4)$.
Count how many ways the first and the last are the same gender.  Break into cases.  Either the first and last are both men, or the first and last are both women.
In either case, pick who the first person is ($8$ ways if woman, $4$ ways if man), followed by picking who the last person is next ($7$ remaining if woman, $3$ remaining if man).  Then pick the second and third persons for the team ($10\cdot 9$ ways regardless).  Apply multiplication rule of counting and addition principle of counting, take the total count and divide by the sample space size, and complete the simplifications to arrive at the answer.
