How is the probability mass function for the negative binomial distribution derived under the total-trials convention?

Conditions

• Trials are independent.
• Success probability p is constant.
• $x \ge r$.

Reasoning, step by step

1. Require the final trial (trial x) to be a success.
2. Require exactly $r-1$ successes in the preceding $x-1$ trials.
3. Calculate the number of ways to choose positions for $r-1$ successes among $x-1$ trials using combinations.
4. Multiply by the probability of r successes and $x-r$ failures.

Example

The formula is $C(x-1, r-1)p^r(1-p)^{x-r}$ for integers $x \ge r$.

Common misconceptions

• Forgetting that the last trial must be a success.
• Using $C(x, r)$ instead of $C(x-1, r-1)$.

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