The Successive Wins Problem

I fortuitously ran into one of my old friend, who instigated my interest in probability puzzles once again. This time not just with a problem but an entire collection of it. So, I decided to devote some time and effort to create a series of blog posts to bring them to you accompanied by their solution. If you are interested about this series follow me to stay tuned. You can also contribute your problems or solutions to this series, by reaching me out on LinkedIn, Twitter, Instagram or Email.

The problem goes like this…
To encourage Elmer’s promising tennis career, his father offers him a prize if he wins (at least) two tennis sets in a row in a three-set series to be played with his father and the club champion alternately: father-champion-father (FCF) or champion-father-champion (CFC), according to Elmer’s choice. The champion is a better player than Elmer’s father. Which series should Elmer choose? Assume that Elmer stops playing when he has satisfied the winning rule or cannot satisfy the winning rule anymore.

Elmer wins the series if he wins two consecutive sets. He can choose in which order to play the series.

People sometimes find the answer and the reason to be quite obvious to this problem.

Elmer’s father is a weaker opponent compared to the club champion. So, no matter what’s the case, playing two times against him will increase Elmer’s chances of winning so he should choose the father-champion-father series. Fair enough, right? The naive approach seems to produce a valid reason!

But the problem with this sort of reasoning is that it assume the total number of games won translates to increased chances of winning ‘at least two sets in a row’! This is somewhat related to assuming the sets can be played independent (But in reality our playing the next set depends on our performance till the sets we have played — If we loose the middle middle set we can’t play anymore, so playing the third set depends on the previous two). Let p1 be the chances of Elmer winning against his dad and p2 be the chances of him winning against the club champion, where p1 ≥ p2. Mathematically, such a thought sprouts from,

The flaw in this argument lies in the very definition of the r.v.s N_CFC and N_FCF, where we are counting individual wins in a single round. But the winning condition is different i.e. ‘at least two sets in a row’. Also summing I_Cs and I_Fs to define N_CFC and N_FCF frees I_C and I_F of the sets previously played!

So, once we rectify this mistake in our mistake we get the following,

E(I_C.I_F) = E(I_C).E(I_F) = p2.p1, as both the indicators are independent and using expectation as bridge between indicator r.v. and probability. The final inequality is trivial as p1 ≥ p2

So, Elmer even though has to face the club champion twice should take that opportunity to have greater chances of bagging in the prize from his dad. As we always do, in the next section, we will try to simulate the scenario and verify our approach.

The Python code for the simulation of the above scenario

import numpy as np

def playSeries(p1, p2):
  # setting opponents probability of winning
  opponents = np.array([p1, p2, p1])
  # simulating the elmer's performance in the games
  elmer_winnings = np.random.rand(3)
  # setting win or lose in each set
  elmer_winnings = elmer_winnings < opponents
  # check if he won the series or not
  if elmer_winnings[0] == 1 and elmer_winnings[1] == 1:
    return 1
  elif elmer_winnings[1] == 1 and elmer_winnings[2] == 1:
    return 1
  else:
    return 0
  
# Simulating FCF Series
N, p1, p2 = 100000, 0.7, 0.2
print(f'Proportion of time Elmer won in {N} FCF simulation: ', np.mean([playSeries(p1, p2) for _ in range(N)]))
print('P[FCF]:', (2-p1)*p1*p2)
print()

# Simulating CFC Series
N, p1, p2 = 100000, 0.2, 0.7
print(f'Proportion of time Elmer won in {N} CFC simulation: ', np.mean([playSeries(p1, p2) for _ in range(N)]))
print('P[CFC]:', (2-p1)*p1*p2)

It gives the following output

Proportion of time Elmer won in 100000 FCF simulation:  0.18391
P[FCF]: 0.182

Proportion of time Elmer won in 100000 CFC simulation:  0.25375
P[CFC]: 0.252

Great! Hope you learnt to be careful in avoiding some basic confusion which arises when looking at a problem through the naive lens. Will be back with more problems from the collection :)