This document discusses using dynamic programming to solve the binomial coefficient and world series problems. It defines binomial coefficient as the number of ways to choose a subset of objects from a larger set. It then describes the world series problem as two teams competing in a series of games until one wins n games. A recursive formula is developed to calculate the probability of different outcomes. Dynamic programming is applied by storing probabilities in an array to speed up the algorithm and solve it in O(4n) time.

1. Binomial Coefficient and World series problem
using Dynamic Programming approach
Submitted by – Samyak Jain (MCA/25014/2022)
Abhishek Ahlawat (MCA/25017/2022) Submitted to – Madhavi Sinha Mam

2. What is Binomial Coefficient?
Binomial Coefficient is used to denote the number of
possible ways to choose a subset of object of a given
number from a larger set.

3. Where did Binomial Coefficient come from?

4. Suppose , we have to find the coefficient of the (x + y)^100 , then it will be very
difficult to find the binomial coefficient using Pascal’s Triangle.
So, Pascal drives the formula to find the binomial coefficient of any (x + y)^n .

5. Binomial Coefficient using Dynamic Programming

8. WORLD SERIES PROBLEM
 Teams A and B compete in a series of games.
• The winner is the first team to win n games.
• The series ends as soon as the winner is decided.
• At most 2n–1 games are played.
 We assume that there are no tied games, that the result of each match are
independent .
 There is a constant chance p that team A will win and hence a constant probability
q = 1 – p that team B would win for every given match.

9. RECURSIVE FORMULA
Given that in a single contest between team A and team B, the probability team A wins is
p,(and the probability that team B wins is 1 – p, so we are precluding ties),what is the
probability in i+j games played, that team A will win i games and team B will win j games.
Let this value be denoted by the function p(i,j).
In order to obtain the dynamic programming solution, we must first develop a
recursive formula for the function p(i,j). In order for team A to have won i games
and team B to have won j games, before the last game, either A won i and B won
j-1 OR A won i-1 and B won j. Here is a recursive formula that captures that
reasoning:
p(i, j) = p(i, j - 1)(1 - p) + p(i -1, j) p

10. WORLD SERIES DYNAMIC PROGRAMMING
P(i, j) is the probability that the team A will win if it gets i more victories
and the team B will win if it gets j more victories.
For example, before the first game off the series the probability that
team A will be the overall winner is P(n, n): both teams still need n
victories.
If team A has already won all the matches it needs, then it is of course
certain that they will win the series:
P(0, i) = 1; 1 ≤ I ≤ n.
Similarly , if team B require 0 victories, then they have already won the
series , and the probability that team A will be overall winner is zero .
P(i,0) = 0 ; 1 ≤ I ≤ n.

11.  Since, team A win any given match with probability p and lose it with
probability q .
 P(i,j) = pP(i-1,j) + qP(i,j-1); i≥1 , j≥1.
 Thus we can compute p(i,j) as follows .
Function P(i,j)
if i = 0 then return 1
else if j = 0 then return 0
else return pP(i-1,j) + qP(i,j-1)

12.  Let T(k) be the time needed in the worst case to calculate P(i, j), where
k = i + j. With this method, we see that
T(k) is therefore in O(2n),
which is O(4n), if i=j=n

13.  To speed up the algorithm , we declare an array of appropriate size and then
fill the entries .


15. REFERENCES
WORLD SERIES PROBLEM
 http://cs360.cs.ua.edu/lectures/37%20Dynamic%20Programming.pdf
 https://cerocks.files.wordpress.com/2011/03/fundamentals-of-algorithmics-
brassard_ingles.pdf
BINOMIAL COEFFICIENT
 https://www.youtube.com/watch?v=GmB0cIY7uMk
 https://www.youtube.com/watch?v=eAMCL-mMmcY

16. Thank you!!!