Demonstration of Role of Combinatorics in Probability

There are 8 people where 3 are girls and 5 are boys (refer to  blog 10 March 2017). We are interested in the probability distribution of no. of top 3 positions occupied by girls. Let X be a random variable denoting no. of top 3 positions occupied by girls X = 0, 1, 2, 3

X= 0 top three positions occupied by 0 girls which means all three positions occupied by 3 boys

X = 1 means one position occupied by girls implying two positions occupied by boys

X = 2 means two positions occupied by girls implying one position occupied by a boy

X = 3 means top three positions occupied by three girls

P(X = 0) = P (Only boys in the top 3) = C(5, 3) C(3, 0) 3!5!/8! = 10/56

C(5, 3) means choosing 3 boys out of 5 boys
C(3, 0) means choosing 0 girls out of 3 girls

3!5! mean arranging these three boys in three positions and remaining 5 students ( girls and boys) in 5 positions.
Similarly

P(X = 1) = P (One girl and two boys) = C(5, 2) C(3, 1) 3!5!/8! = 30/56

P(X = 2) = P( two girls and one boy) = C(3, 2) C(5, 1) 3!5!/8! = 15/56

P(X = 3) = C(3, 3) 3!5!/8! = 1/56

Total probability is 1

P(X = 0) + P(X=1) + P(X=2) + P(X = 3) = 1

This probability distribution can also be used in describing the gender distribution among deaths of first 3 people out of a total of 8 people. Here permutation and combination has been used together as a counting technique for selection (where order is not important) and arrangement (where order is important).