In this Presentation you get a brief knowledge about World Series Problem.

1. World Series Problem : using
dynamic programming approach
Abhinav Kumar Singh

2. Introduction
 The World Series is a seven-game series that terminates as soon as
either team wins four games.
 The World Series problem allowed us to see the power of dynamic
programming to compute a result iteratively from very simple rules,
filling a table of partial results that leads to a complete answer.

3. THE WORLD SERIES PROBLEM
 Consider a tournament in which two teams A and B compete in no more than
2n -1 games, with the winner being the first team to win n times. We assume
that there are no tied games, that the results of each match are independent,
and that 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.

4. Implementation
 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).

5. Implementation
 An alternative way to solve the problem involves dynamic
programming. 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

6. Now, instead of actually using recursion in our code to figure this out, we can
convert our algorithm into dynamic programming by creating a two-dimensional
array that stores the answers to all necessary recursive calls.

7. Algorithm (Code)
public static int WS(int i, int j, double p)
{
double[][] prob = new double[i+1][j+1];
prob[0][0] = 1;
// Fill in probabilities for team A sweeping. for (int Awin=1; Awin<=i; Awin++) prob[Awin][0] =
p*prob[Awin-1][0];
// Probabilities for team B sweeping for (int Bwin=1; Bwin<=j; Bwin++) prob[0][Bwin] = (1-
p)*prob[0][Bwin-1];
// Here's where we fill in the table, using the recursive
// formula.
for (int Awin=1; Awin<=i; Awin++)
for (int Bwin=1; Bwin<=j; Bwin++)
prob[Awin][Bwin] = p*prob[Awin-1][Bwin] + (1-p)*prob[Awin][Bwin-1];
// Here's our answer. return prob[Awin][Bwin]; }

8. The World Series
Example of Dynamic Programming
 The teams A and B play in not more than 2n- 1 games. assume
that
 There are no tied games
 the probability of winning by team A is p
 the team B wins with q = 1 – p.
 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.

9. The World Series…
 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, P(i, 0) = 0, 1 ≤ i ≤n.
 Finally, since team A wins any given match with probability p and
loses it with probability q,P(i, j) = ?

10. The World Series…
 P(i; j)=pP(i - 1; j) + qP(i; j-1); i≥1;j≥1
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)
 Let T(k) be the time required to compute P(i; j) where k = i + j T(1) =
c
T(k) ≤2T(k; 1)+d; k > 1

11. The World Series…
 T(k) ≤ 4T(k-2) + 2d+d; k > 2
...
≤2k-¹T(1) +(2k-2 +2k-3+...+2+ 1)d
= 2k-1c +(2k-1-1)d
= 2k(c/2+d/2) – d
→ T(k) € 0(2)

13. The World Series…
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)

15. Example
 Dodgers and Yankees are playing the World Series in which either team needs to win n games
first. (number of maximum game=2n-1)
 - Suppose that each team has a 50% chance of winning any particular game. - Let P(i, j) be the
probability that if Dodgers needs i games to win, and Yankees needs j games, Dodgers will
eventually win the Series.
 - Ex: P(2, 3) = 11/16
 - Compute P(i, j) (0 <= i, j <= n) for an arbitrary n.
 P(0,1)=1(Win), Yankees W
 P(0,2)=1(Win), Yankees WW
 …
 P(1,0)=0(Lose), Yankees L
 P(2,0)=0(Lose), Yankees LL
 P(2,2)

16. THANK YOU