# Probability Analysis

<figure><img src="/files/2QtOwy5VWRD0VgACWdXV" alt=""><figcaption></figcaption></figure>

*The probability of an attacker catching up from a given deficit is analogous to a Gambler's Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as follows \[8]:*

*p = probability an honest node finds the next block q = probability the attacker finds the next block qz = probability the attacker will ever catch up from z blocks behind*

*qz = { 1 if p≤q } { (q/p)z if p>q }*

Satoshi Nakamoto, Bitcoin Whitepaper

Because of the pseudo-random processes involved, the probability of an attacker regaining the lead is very strongly related to their **own ability to perform proof-of-work**. In a situation where they can perform proof-of-work **equal to or greater than the rest of the network**, there is a 100% chance they will eventually have the leading chaintip, however **if their capability is less than the network**, the chance of them ever catching up goes down with the square of the gap between the two chaintips, leading them to an ever increasing deficit against the rest of the network.

In simple terms, this mathematical framework illustrates that **as the number of blocks the attacker must catch up with increases**, the **probability of success diminishes exponentially**, particularly when the honest chain maintains a lead.


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://hub.bsvblockchain.org/higher-learning/bsv-academy/bitcoin-whitepaper-series/section-11-calculations-bitcoin-whitepaper/probability-analysis.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.