The chances of you existing:

If you go back 10 generations (250 years) the chance of you being born at all is  at most 1 divided by 6 x 10100 or
1 in 60000000000000000000000000000000000 00000000000000000000000000 000000000000000000000000000000000000.
In gambling, even a chance of 1 to 100 is not worth a gamble.
Are you ever lucky to be alive!!

If you go back 1 million years or 40 000 generations (each generation is considered 25 years), your chance of being born is at most 1 in 1.8 x 10403167 or
18 with 403,166 zeros after the 1.

In other words your chances of existing is essentially zilch, even if we were considering this possibility only a short 250 years ago.
Right now you do exist, so the actual chance is 100%, but the predicted likelihood in the past of you being born would be essentially zero.

The calculations of  these numbers (calculations shown below) tend to overestimate your chances of being.
They only include numbers based on your father's contribution, his father's contribution, his father contribution, and so on.  In other words the calculations are based only on the direct male line.
They do NOT include the following factors that would make the chances of your existence even less:
- the men and women that died in war, famine, disease without or with fewer offspring-- in some generations 50% of humanity.
- the men and women that died from natural causes.
- children that died before reaching the age of reproduction.
- fetuses and fertilized eggs that died and were naturally aborted.

You are the result of many generations of survivors.  One of the people that died prematurely could have been your Dad or your Dad's Dad and so on.  Somehow, none of your forefathers died before passing on his genes to the next man in your lineage.  Because of the deaths mentioned most human lineages died out, while luckily yours survived.  Lo and behold, here you are!
Because of these deaths, your odds of existing were actually much less than the odds given below.

The Calculations:
As mentioned above, these calculations assume only your father, his father, his father, and so on.  The number of sperm produced by your father is astronomical compared to the number of eggs produced by your mother, so this is a reasonable procedure to show how slight your existence chances were.
Also I am assuming that each separate reproductive generation consists of 25 years

Some background statistics based on the book "Sperm Wars" by Robin Baker,  University of Manchester.
On average a man between puberty (I suggest age 15 y) and 30 years produces 300 million sperm/day (300,000,000).
On average a man between 30 y and 50 years produces 230 million sperm/day (230,000,000)
On average a man between 50 and 75 years produces 50 million sperm/day      (50,000,000)

The last two sperm/day numbers were calculated as follows:
At age 50 years the average man produces 175 million/day -- so averaging 300 million/day and 175 million/day you get 230 million/day.
At age 70 years the average man produces 20 million/day -- so I could average 175 million and 20 million, but to be on the safe side I decreased it to 50 million/day (keeping my results on the conservative side -- I will round lower for the calculations that follow).
Each sperm has a unique set of DNA.
These sperm will appear in intercourse,  masturbation, nocturnal emission.  They all have to go somewhere.

On average 2 to 3 offspring survive that will reproduce.  This average seems to be fairly constant in modern societies, agricultural societies and in hunter-gatherer societies.  So, for all generations it hovers around 2 to 3 reproducing survivors.  On average, 2 to 3 sperm will result in the fertilization of eggs that results in 2 or 3 reproducing individuals -- the rest of the sperm are doomed.

Kind of sperm:
An average man produces 3 kind of sperm:
(1) Killer Sperm constitute about 83% of all the sperm -- these are sperm that attack any sperm from another man that may be in the woman. These sperm carry a lethal dose of poison in their head, seek out foreign sperm and inject it with this poison, killing it.  Apparently after a few injections the Killer Sperm has spent all it's energy and also dies.

(2) Blocker Sperm constitute about 16% of all the sperm -- these try to block entrances and storage places so that any sperm from another man cannot reach the woman's egg.

(3) Egg Getter Sperm constitute about1% of all the sperm -- these are the sperm that can possibly fertilize the woman's egg if they ever get there.

So 99% of sperm make war on other men's sperm to  annihilate them so that they won't fertilize the woman's egg, and 1% have the potential to fertilize the woman's egg.

In our calculations we will only use the numbers for the Egg Getter Sperm.

The Actual numbers:
(A) The number of Egg Getter Sperm produced by an average man in his lifetime:

In the following calculations I will round off the numbers lower to be even more conservative.
(1) Between puberty (15 years old) and 30 years:
At a rate of 1% of 300,000,000 = 3,000,000 Egg Getter Sperm/day
15 years has 5475 days.
Therefore, on average a man will produce 5475 days x 3,000,000 Egg Getter Sperm/day = 1.6 x 1010 Egg Getter Sperm during this time (16,000,000,000).

(2) Between 30 years and 50 years:
At a rate of 1% of 230,000,000 = 2,300,000 Egg Getter Sperm/day
20 years has 7300 days.
Therefore, on average a man will produce 7300 days x 2,300,000 Egg Getter Sperm/day = 1.6 x 1010 Egg Getter Sperm during this time (16,000,000,000).

(3) Between 50 years and 75 years:
At a rate of 1% of 50,000,000 = 500,000 Egg Getter Sperm/day
25 years has 9125 days.
Therefore, on average a man will produce 9125 days x 500,000 Egg Getter Sperm/day = 4 x 109 Egg Getter Sperm during this time (4,000,000,000).

On average, in a man's lifetime he produces 16,000,000,000 + 16,000,000,000 + 4,000,000,000 = 36,000,000,000 Egg Getter Sperm for a possible 36,000,000,000 unique offspring (3.6 x 1010).

(B) Fraction of Egg Getter Sperm that result in offspring that will reproduce:

Since an average man will have 2 to 3 surviving offspring (let's take 3 to be on the conservative side) the fraction of all the Egg Getter Sperm that will result in another person is:  3 x 1 / 3.6 x 1010
or 1 / 1.2 x 1010
or 1 / 12,000,000,000.
Let's be even more conservative and take this fraction to be your probability of being instead of the probability of 3 survivors.
In other words of all the Egg Getter Sperm your father produces, the chances are that the sperm with your name on it that will actually become you is 1 to 12,000,000,000 -- not great odds!

1 /12,000,000,000 x 1 / 12,000,000,000 = 1 / 144,000,000,000,000,000,000
or 1 to 144,000,000,000,000,000,000
or 1 / 1.44 x 1020.

If you take 10 generations (10 generations x 25 years/ generation = 250 years) you must multiply 1 / 12,000,000,000
ten times, or (1 / 1.2 x 1010)10 = 1 / 6 x 10100
or 1 to 6 x 10100.  This is the number given at the beginning of this page for 10 generations.

If you take 1 million years of evolution (actually one should use many millions of years, but to be conservative we will limit our calculation to 1 million years):
40,000 generations
(1 / 1.2 x 1010)40000 = 1 / 1.8 x 10403167 chance that you should exist. (a lot of wasted sperm, eh?)

This is  mind boggling odds of 1 to 18 with 403,166 zero's after it.  In other words your chances of being are essentially zero.

(C) Try to visualize these numbers:

The age of the universe since the Big Bang is 13.7 billion years.

In seconds: 13.7 x 109 year x 365 day/year x 24 h/day x 3600 seconds/hour = 4.32 x 1016 seconds
or 43,200,000,000,000,000 seconds is the age of the Universe -- a very, very long time!!

If 1 Egg Getter Sperm were produced every second of the lifetime of the universe then it would take:
6 x 10100 / 4.32 x 1016 = 1.4 x 10 89 universe lifetimes to definitely result in you becoming, that is, if you calculate only 10 generations back (250 years).
14 with 88 zeros after it of universe lifetimes -- essentially an infinite time!

If you go back 1 million years it would take 42 x 10403149 universe lifetimes to definitely result in you.
42 with 403,149 zeros after the 42 of it lifetimes of the Universe!

All these numbers err on the side of caution -- a lot of caution.

(D) Aren't You lucky?

If you consider almost any time in the past and try to calculate the probability of you coming into being, even 2 generations ago, your possible existence is so low that it's almost not worth thinking about.
Aren't you happy you're alive???  Lucky YOU!