Lesson goal: Tournament odds and binomial series

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In the 1971 MAA High School Mathematics Contest (Problem #23), contestants tackled this classic championship puzzle:

Teams A and B are playing a series of games. If the odds for either team to win any single game are even ($50\% - 50\%$), and Team A must win 2 games or Team B must win 3 games to win the series, what are the odds favoring Team A to win the championship?

At first glance, it might seem like a simple ratio of $3 \text{ to } 2$. But in playoff tournaments (like the World Series or NBA Finals), games are independent trials, and probability compounds multiplicatively!

To analyze the problem:
  • The tournament must finish in at most $2 + 3 - 1 = 4$ games. By the end of 4 games, it is mathematically impossible for neither team to have reached their target!
  • Each 4-game sequence has $2^4 = 16$ equally likely outcomes.
  • Team B can only win if they win 3 or 4 games:
    • Winning all 4 games: $\text{BBBB}$ ($1$ way)
    • Winning 3 of 4 games: $\text{ABBB}, \text{BABB}, \text{BBAB}, \text{BBBA}$ ($4$ ways)
  • Total ways Team B wins: $1 + 4 = 5$ ways out of 16 ($31.25\%$).
  • Total ways Team A wins: $16 - 5 = 11$ ways out of 16 ($68.75\%$).
Therefore, the odds favoring Team A are exactly 11 to 5!

In this lesson, we will use both a Monte Carlo simulation (simulating 10,000 tournaments with random coin flips) and an exhaustive tree to prove the result.
a_wins = 0
b_wins = 0
for sim = 1, 10000 do
a = 0; b = 0
while a < 2 and b < 3 do
if math.random() < 0.5 then a = a + 1 else b = b + 1 end
end
if a == 2 then a_wins = a_wins + 1 else b_wins = b_wins + 1 end
end
Move the mouse over a dotted box for more information.

Notice how close the simulated win ratio gets to the exact theoretical probability: $$\frac{11}{16} = 0.6875 = 68.75\%$$ $$\text{Odds Ratio} = \frac{11}{5} = 2.20$$

Now you try. Run the simulation with num_sims = 50000 for even higher precision. Then try changing Team A's required wins to 3 (a standard best-of-5 series) to see how the odds become perfectly 50-50!

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