Last Updated on April 1, 2026 by Rajeev Bagra

In a biased random walk, you:

• Move up (+1) with probability p
• Move down (−1) with probability q
• And q > p (so the game is against you)

A powerful result says:

👉 The probability of ever reaching +a starting from 0 is:

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🧠 The Core Idea

Think of the process as a sequence of levels:

0 → 1 → 2 → … → a

To reach +a, you must successfully move through each level.

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🔁 Step 1: Probability of Reaching +1

Even reaching +1 is not guaranteed because:

• You may go down first
• Then struggle to come back

From recurrence analysis:

👉 Probability of ever reaching +1:

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🔁 Step 2: Moving from +1 to +2

Once you reach +1:

• The process “restarts”
• Past history doesn’t matter (Markov property)

👉 Probability of going from +1 to +2:

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🔗 Step 3: Chain the Events

To reach +2:

• First reach +1 →
• Then reach +2 →

Multiply:

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🔁 Step 4: General Case

To reach +a:

• You need a successful upward transitions

Each has probability:

So:

👉 Total probability:

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📉 Why the Probability Shrinks

Since:

So:

• 
• 

👉 The probability decays exponentially

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📊 Example

If:

Then:

So:

• Reach +1 →
• Reach +2 →
• Reach +3 →

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💡 Real-World Insight

When every step is biased against you, success probability shrinks exponentially with distance.

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🧠 One-Line Takeaway

👉 “To reach +a, you must win a biased game a times in a row, which gives:

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