Quote:Ace2If you start rolling a standard die and record the results, how many rolls will it take, on average, for all six numbers to have an equal number of hits?

Infinity.

Even with just a coin, the expected number of flips for the number heads and tails to be equal is infinity.

That paradoxical thing is that the probability they will ever be the same is 1. If it will eventually happen, how can it take an infinite number of flips?

Quote:DJTeddyBearIt takes 25,855 spins before it exceeds Excel's capabilities and incorrectly returns a value of zero.

There is a way around it.

Note that log ((36/37)^N) = N (log 36 - log 37), where log is the base 10 logarithm

If the value = a + b, where a is an integer and 0 <= b < 1, then (36/37)^N = 10^a 10^b

You can calculate a and b using the floor() function

26,000 | 4.170 558 E-310 |

27,000 | 5.259 820 E-322 |

28,000 | 6.633 574 E-334 |

29,000 | 8.366 124 E-346 |

30,000 | 1.055 118 E-357 |

40,000 | 1.074 158 E-476 |

50,000 | 1.093 541 E-595 |

60,000 | 1.113 274 E-714 |

70,000 | 1.133 363 E-833 |

80,000 | 1.153 815 E-952 |

90,000 | 1.174 635 E-1071 |

100,000 | 1.195 832 E-1190 |

200,000 | 1.430 013 E-2380 |

300,000 | 1.710 055 E-3570 |

400,000 | 2.044 938 E-4760 |

500,000 | 2.445 401 E-5950 |

600,000 | 2.924 288 E-7140 |

700,000 | 3.496 956 E-8330 |

800,000 | 4.181 771 E-9520 |

900,000 | 5.000 694 E-10,710 |

1,000,000 | 5.979 988 E-11,900 |

2,000,000 | 3.576 025 E-23,799 |

3,000,000 | 2.138 459 E-35,698 |

4,000,000 | 1.278 796 E-47,597 |

5,000,000 | 7.647 183 E-59,497 |

6,000,000 | 4.573 006 E-71,396 |

7,000,000 | 2.734 652 E-83,295 |

8,000,000 | 1.635 318 E-95,194 |

9,000,000 | 9.779 184 E-107,094 |

10,000,000 | 5.847 940 E-118,993 |

You refreshed my memory.Quote:WizardInfinity.

Even with just a coin, the expected number of flips for the number heads and tails to be equal is infinity.

That paradoxical thing is that the probability they will ever be the same is 1. If it will eventually happen, how can it take an infinite number of flips?

I’ve seen the case for a random walk where there’s a 50/50 chance of going left or right. It can be proven that there’s a 100% chance of eventually returning to the origin, but potentially infinite waiting time. Same concept as the coin flip..

Thanks for the replies

Quote:Ace2You refreshed my memory.

I’ve seen the case for a random walk where there’s a 50/50 chance of going left or right. It can be proven that there’s a 100% chance of eventually returning to the origin, but potentially infinite waiting time. Same concept as the coin flip..

Thanks for the replies

You're welcome.

Funny how many paradoxes come up when it comes to infinity.

I still say infinity is more of a philosophical concept than a number and claim there is nothing in the physical universe that is infinite in nature.

So you believe space and time come to an end somewhere/sometime?Quote:Wizard

I still say infinity is more of a philosophical concept than a number and claim there is nothing in the physical universe that is infinite in nature.

Much of calculus is based on infinity

Quote:Ace2So you believe space and time come to an end somewhere/sometime?

I believe space does. Time is a tougher one.

Quote:Much of calculus is based on infinity

I, of course, know that. However, calculus is not a physical thing.

Quote:Ace2So you believe space and time come to an end somewhere/sometime?

I believe space does. Time is a tougher one.

Quote:Much of calculus is based on infinity

I, of course, know that. However, calculus is not a physical thing.

Quote:Ace2So you believe space and time come to an end somewhere/sometime?

I believe space does. Time is a tougher one.

Quote:Much of calculus is based on infinity

I, of course, know that. However, calculus is not a physical thing.

Quote:Ace2So you believe space and time come to an end somewhere/sometime?

I believe space does. Time is a tougher one.

Quote:Much of calculus is based on infinity

I, of course, know that. However, calculus is not a physical thing.

I’m barely an arithmetic guy. Calculus? Fuggedanoudit.Quote:ThatDonGuyThere is a way around it.

Note that log ((36/37)^N) = N (log 36 - log 37), where log is the base 10 logarithm

If the value = a + b, where a is an integer and 0 <= b < 1, then (36/37)^N = 10^a 10^b

You can calculate a and b using the floor() function

On the other hand...

I’m a little surprised you didn’t continue this till the exponent got too big for Excel - or became got expressed as scientific notation itself.Quote:

10,000,000 5.847 940 E-118,993