Theorem(Erd6s 1963) If ()(1-)<1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Fixed any S∈ . Event As:no player in V\beat all players in S. PrAs=(1-2k)”-
If n k ⇥ 1 2k⇥nk < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Fixed any S [n] k ⇥ Event AS : no player in V \S beat all players in S. Pr[AS] = 1 2k⇥nk
Theorem(Erd6s 1963) If()(1-2-k)”- 1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Event As:no player in \S beat all players in S. Pr[As=(1-2k)”-k Pr ≤∑(1-2k)-k <1 s∈() S∈()
If n k ⇥ 1 2k⇥nk < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Event AS : no player in V \S beat all players in S. Pr[AS] = 1 2k⇥nk Pr < 1 ⇧ ⇤ ⌥ S( [n] k ) AS ⇥ ⌃ ⌅ ⇥ S⇥( [n] k ) (1 2k) nk
Theorem(Erd6s 1963) If ()(1-2-k)"<1 thon there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Event As no player in \S beat all players in S. V<1 se() 0 S∈()
If n k ⇥ 1 2k⇥nk < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Event AS : no player in V \S beat all players in S. Pr < 1 ⇧ ⇤ ⌥ S( [n] k ) AS ⇥ ⌃ ⌅ Pr[T is k-paradoxical] = 1 Pr ⇧ ⇤ ⌥ S( [n] k ) AS ⇥ ⌃ ⌅ > 0
Theorem(Erdos 1963) If (R)(1-2-k)"<1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. PrT is k-paradoxical>0 There is a k-paradoxical tournament on n players
If n k ⇥ 1 2k⇥nk < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Pr[T is k-paradoxical] > 0 There is a k-paradoxical tournament on n players
The probabilistic Method Pick random ball from a box, 00 Pr[the ball is blue]>0 000 →There is a blue ball. Define a probability space O,and a property P: Pr[P(x)]>0 X >xE with the property P
The Probabilistic Method • Pick random ball from a box, Pr[the ball is blue]>0. ⇒ There is a blue ball. • Define a probability space Ω, and a property P: Pr x [P(x)] > 0 = ⇤x ⇥ with the property P