Editing Birthday problem (section)

===Reverse problem===
The reverse problem is to find, for a fixed probability {{mvar|p}},
the greatest {{mvar|n}} for which the probability {{math|''p''(''n'')}} is smaller than the given {{mvar|p}}, or the smallest {{mvar|n}} for which the probability {{math|''p''(''n'')}} is greater than the given {{mvar|p}}.{{citation needed|date=September 2019}}

Taking the above formula for {{math|''d''&nbsp;{{=}}&nbsp;365}}, one has

:<math>n(p;365)\approx \sqrt{730\ln\left(\frac{1}{1-p}\right)}.</math>

The following table gives some sample calculations.
:{| class="wikitable"
|-----
! {{mvar|p}} || {{mvar|n}}
! {{math|''n''↓}} || {{math|''p''(''n''↓)}} || {{math|''n''↑}} || {{math|''p''(''n''↑)}}
|-----
| <span style="color:maroon">0.01</span>
| 0.14178{{sqrt|365}} = <span style="color:maroon">2.70864</span>
| align="right" | 2 || 0.00274 || align="right" | 3
| <span style="color:maroon">0.00820</span>
|-----
| 0.05 || 0.32029{{sqrt|365}} = 6.11916
| align="right" | 6 || 0.04046 || align="right" | 7 || 0.05624
|-----
| <span style="color:maroon">0.1</span>
| 0.45904{{sqrt|365}} = <span style="color:maroon"> 8.77002</span>
| align="right" | 8 || 0.07434 || align="right" | 9
| <span style="color:maroon">0.09462</span>
|-----
| <span style="color:maroon">0.2</span>
| 0.66805{{sqrt|365}} = <span style="color:maroon">12.76302</span>
| align="right" | 12 || 0.16702 || align="right" | 13
| <span style="color:maroon">0.19441</span>
|-----
| 0.3 || 0.84460{{sqrt|365}} = 16.13607
| align="right" | 16 || 0.28360 || align="right" | 17 || 0.31501
|-----
| 0.5 || 1.17741{{sqrt|365}} = 22.49439
| align="right" | 22 || 0.47570 || align="right" | 23 || 0.50730
|-----
| 0.7 || 1.55176{{sqrt|365}} = 29.64625
| align="right" | 29 || 0.68097 || align="right" | 30 || 0.70632
|-----
| 0.8 || 1.79412{{sqrt|365}} = 34.27666
| align="right" | 34 || 0.79532 || align="right" | 35 || 0.81438
|-----
| 0.9 || 2.14597{{sqrt|365}} = 40.99862
| align="right" | 40 || 0.89123 || align="right" | 41 || 0.90315
|-----
| 0.95 || 2.44775{{sqrt|365}} = 46.76414
| align="right" | 46 || 0.94825 || align="right" | 47 || 0.95477
|-----
| <span style="color:maroon">0.99</span>
| 3.03485{{sqrt|365}} = <span style="color:maroon">57.98081</span>
| align="right" | 57
| <span style="color:maroon">0.99012</span>
| align="right" | 58 || 0.99166
|}
Some values falling outside the bounds have been <span style="color:maroon">colored</span> to show that the approximation is not always exact.