Last Updated on May 6, 2026 by Rajeev Bagra

🧩 Initial Context

Calvin and Hobbes are playing a match.

• Calvin wins each game with probability p
• Hobbes wins with probability q = 1 − p
• The match continues until one player leads by 2 games

We define:

During the solution, an important question arises:

👉 Why do we include both

and

when Calvin ultimately needs to move toward +1 to win?

❓Q1: What does state “+1” mean?

✅ Answer

It means Calvin is currently ahead by 1 game.

Example:

• Calvin: 3
• Hobbes: 2

Difference:

❓Q2: What does state “−1” mean?

✅ Answer

It means Hobbes is ahead by 1 game.

Example:

• Calvin: 2
• Hobbes: 3

Difference:

❓Q3: Why do we include both +1 and −1 states?

✅ Answer

Because from the starting tie, both states are possible after the next game.

From tie:

Tie
├── Calvin wins → +1
└── Hobbes wins → -1

So total probability must include both branches.

❓Q4: Why isn’t focusing only on +1 enough?

✅ Answer

Because Calvin can still recover from −1.

From −1:

-1
├── Calvin wins → back to tie
└── Hobbes wins → lose

So the −1 branch still contributes to Calvin’s overall winning probability.

❓Q5: What is the equation for total probability?

✅ Answer

This means:

👉 Overall chance =
(probability entering a state) × (chance of eventually winning from that state)

❓Q6: Why is

✅ Answer

From −1:

• Calvin must first win next game (probability p)
• That returns the game to tie
• From tie, his winning chance is x again

So:

❓Q7: Why does the game “reuse” x repeatedly?

✅ Answer

Because whenever the game returns to tie, we are back to the exact same starting situation.

This is called a recursive probability structure.

❓Q8: What is the key intuition of this problem?

✅ Answer

👉 The game behaves like:

“Win now, or return to the same state and try again.”

That is why recursive equations naturally appear.

🧠 Key Learning Takeaways

• Model problems using states
• Use conditioning on the next step
• Include all possible branches
• Repeated states create recursion

🎯 Final Insight

Even though Calvin ultimately needs to move toward +1 and +2 to win, the −1 state still matters because Calvin can recover from it and continue the match.