her definition where the set of trials starts
            at one, sometimes named <span class="emphasis"><em>shifted</em></span>).
          </p></td></tr>
</table></div>
<p>
          The geometric distribution assumes that success_fraction <span class="emphasis"><em>p</em></span>
          is fixed for all <span class="emphasis"><em>k</em></span> trials.
        </p>
<p>
          The probability that there are <span class="emphasis"><em>k</em></span> failures before the
          first success is
       