What percentage of the population has the “magic gene”?

Question

So I have a world that has magic. Whether you can use magic is hereditary, and there's about a 1/4 chance of inheriting the "magic gene".

What I want to know is, will the percentage of magic users stay roughly the same, or will it change? And if it stays the same, will the percentage be near 25%, as there is a 1/4 chance of inheriting it, or will the population be different?

If it's relevant, the people on this world have two semi-closed off societies. One is of magic-users and one isn't, and at birth you generally get shipped off to the one that you are "supposed" to be in.

EDIT: The gene is a recessive gene like the classic detached vs attached earlobes example in genetics.

Answer 1

This depends on whether the magic using gene gives you a competitive advantage.

If it does then you will tend to have more magic users breed and they will also become more likely to pass it on to their offspring.

In other words unless it also comes with a "penalty" you will end up with most of the population having it after time.

One way to prevent this would be to have it work a bit like sickle cell anemia. Having a few sets of "magic genes" is great and gives you magic. Having too many though gives you progressively worse and worse side effects.

This would keep the pressure balanced to give you an equilibrium between some people being magic but not so many people having the magic gene that their heads explode (or whatever happens that prevents you breeding when you have too many).

Answer 2

The percentage having a single copy of the gene will shrink

If it's relevant, the people on this world have two semi-closed off societies. One is of magic-users and one isn't, and at birth you generally get shipped off to the one that you are "supposed" to be in.

Because of this, all people with two copies of the gene are in one society (magic-users). The other society has a mix of people with one and zero copies of the gene. Let's assume that both societies have the same number of children per woman (and that women and men are equally likely to have the gene). As a simplifying assumption, there are no cross-society marriages or misclassified members.

25% magic-users

If everyone who has the gene has two copies, then this will be stable. Magic-users only breed with magic-users. Non-magic-users only breed with non-magic-users.

8% magic-users

75% without the gene. 17% with a single copy.

The 8% magic-users add an additional .78% in the next generation. This is based on there being a $17/92 * 17/92$ chance that both parents have a single copy of the gene in the non-magic-user population and a 25% chance that the child will have two copies of the gene.

In the next generation, there are two ways to get single gene children. First, two single gene parents will have a single gene child 50% of the time. That's 1.56%. Second, there is a 50% chance that a single gene parent and a parent without the gene will have a single gene child. That's 13.9%. The total then is 15.5%. So the total with one or more copies of the gene is 24.3%.

Arbitrary percentage

For an arbitrary percentage $p$ of the population having a single copy of the gene.

Chance of a magic-using child from the non-magic-using population:

$$ p^2 * .25 $$

Chance of a carrier (single copy of the gene) from the non-magic-using population.

$$ p^2 * .5 + p * (1 - p) * 2 * .5 = p^2 * .5 + p * (1 - p) = p - .5 * p^2 $$

Note that in the next generation,

$$ p - .5 * p^2 + .25 * p^2 = p - .25 * p^2 < p$$ $$ \forall p > 0 $$

So that we can say that in these circumstances, the chance of having a copy of the gene will fall every generation.

So, not stable unless

From this, we can say that the percentage of people carrying the magic-using gene will fall in each generation until it is eliminated from the non-magic-using population. At that point, the magic-using population will become stable at whatever percentage.

Occasionally mixing the two societies will help this, but I don't believe that it eliminates the basic issue. Removing the gene from one population will cause a steady deterioration until it is eliminated. Note that it may take a long time to get to zero. When the chance of a non-magic-user being a recessive carrier of the gene gets low, then there is little chance of the child having two carrier parents. So with occasional mixing, a practical chance might be something like 1%.

I think that starting from 8% and 17%, you would end up around 12% and 1%. But I'm not sure of my convergence math.

Do non-magic-users have as many kids as magic-users?

Note that all this assumes that there isn't any reason that magic-users would have more children than non-magic-users. For example, if there is some reason why non-magic-users would prefer magic-users as sexual partners, this might not be true.

Answer 3

If magic users have the same number of children as non-magic users:

It will remain 25%

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If magic users produce fewer children because of magic type commitments:

It will be lower than 25%

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If magic users produce more children because of magic type charms:

It will be higher than 25%

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If no one were to train the magic users and everyone walked around not casting fireballs (except LARPers), then for a sufficiently large population, the carriers will very closely match the probability of inheriting the gene. That is:

It will remain 25%

Answer 4

The percentage of magicians will grow.

Per the OP, the magic society and the non-magic society are separate. We can assume the birth rates are identical.

Magic parents only have the recessive "magic" gene, so they can only have magic babies. Therefore, the magic society will NEVER ship off any babies to the non-magic society.

On the other hand, 2 non-magic parents can have magical babies. Therefore, they will regularly ship off magical babies to the magic society.

Since the birth rates in the 2 societies are the same, but the non-magic society is transferring up to 1/4 of its offspring to the magic society, the percentage of magic people in the entire population will grow and grow.

Answer 5

If the 25% chance is based on population percent with the gene...

If the magic gene offers an advantage, the percentage will raise over time. If the magic gene offers a disadvantage, it will decrease. If the magic gene offers a net balance, nothing will necessarily happen either way; other variables will determine the course history takes.

...Oh? But the two segments of society are separated?

Let's say there are 1 million people. Exactly 25% (250,000) are Mages. 75% (750,000) are Mundanes. Let's use a Population Calculator to see what the percentage will look like in 100 years, assuming a 1.2% growth rate for both.

Mages: 824,121 (24.999992416%)

Mundanes: 2,472,364 (75.000007583%)

TOTAL: 3,296,485 (100%)

As you can see, unless an outside force affects that growth percentage, it will maintain 25/75 split. (I even tested for 1000 years later and it was still the same, I'm just sparing you the monstrous numbers.)

That said, this is a 25% chance a person will be born into the magical society, not a 25% chance that they will be born with the Mage genes. There's a clear distinction, but this does allow for the first to hold true... mostly. There's another caveat I didn't include: This assumes that the population of Mundanes don't have the Mage gene at all. If they do have the Mage gene, then the odds are going to be different based on who is breeding with who, as you can see in the next section.

If the 25% chance is based on genetics...

This only holds true if a mundane with the recessive magic gene breeds with another mundane with the recessive magic gene.

As Mundanes breed together, there's a 50% chance that a given Mundane has the gene and a 50% chance they don't. There's a 25% chance that both Mundanes have the gene. If one has the gene, but the other doesn't, there's a 100% chance the child will be born Mundane. If both have the gene, there's a 75% chance the child will be born mundane and 25% chance the child will be born a mage. So, let's assume that two random Mundanes breed, assuming a truly equal distribution of Mundanes with and without the Mage gene. There would be a 6.25% chance that two Mundanes would have a Mage child.

But Sora, where are you getting 6.25% from?

When you're calculating the odds of this event happening, you have to calculate it by looking at the likelihood of the events happening separately and then combining them. Let me clarify. There's a 50% chance that a given Mundane will have the gene. Since we need two Mundanes, that means we would consider the likelihood twice: 50% of 50% is 25%. So, with that, we now know there's a 25% chance that two Mundanes that both have the recessive gene will breed.

If you look at the following Punnett Square, you can see the odds of a Mage (mm) being born from 2 Mundanes with the Mage gene is 1 out of 4 or 25%.

Since we're trying to see the odds of getting a mage from 2 random Mundanes, we know the odds of 2 Mundanes both having the right gene is 25% and now we know that the odds of a Mage being born from them is also 25%. So, we have to find out what 25% of 25% is: 1/16 or 6.25%.

If two Mages breed together or a Mundane without the recessive gene breeds with anyone, (Mage, Mundane, it doesn't matter,) you're guaranteed 100% outcome of Mage and 100% outcome of a Mundane, respectively.

below: The odds of getting a Mage from 2 Mages... 100%

below: The odds of getting a Mage from a Mage and a Mundane without the Mage gene... 0%

above: The odds of getting a Mage from a Mundane with and a Mundane without the Mage gene... 0%

So, let's say we want to see what the odds are of a Mage being born under the following assumptions:

1. Equal Mage and Mundane Population
2. Equal chance a Mundane does or does NOT have the Mage gene
3. Mage gene is recessive
4. Societies are NOT segregated
5. There are no social barriers barring relationships
6. There is no genetic reason to prefer Mundane or Mage over the other
7. Our sample is all equally likely to be involved with someone else from the group

Well to test the odds of this, we have 4 males and 4 females. 2 males and 2 females are Mages. 1 male and 1 female are Mundanes with the Mage gene. 1 male and 1 female are Mundanes without the Mage gene. The outcomes we are testing for will be as follows:

1. Mundane born
   • a) Mundane with Mage gene born
   • b) Mundane without Mage gene born
2. Mage born

First, let's check the odds of an individual male winding up with an individual female... (or vice versa. The order here is irrelevant.) That means each male has a 25% chance of picking any given female (assuming no conflicts and fighting over choice of mate occur).

There's a 50% chance the male is a Mage and a 50% he's not a Mage.

Let's assume we grab one of the 2 male Mages. There's a 50% chance his mate will be a Mage and 50% chance she won't be a Mage. That brings the odds of a Mage being born to become 25%. We've been over this already.

But what about that 50% chance his mate is NOT a Mage? As we saw, if the female Mundane does NOT have the Mage gene, their children will NOT be born mages, but they will all have the Mage gene. If the female Mundane does have the Mage gene, we get something different.

As you can see, there is now a 50% chance that the child will be born as a Mage or Mundane. I feel like you get the general idea, so instead of repeating the same song and dance, let's look at these percents so far:

Odds Checker

A: Pairing - Odds of Each Pairing

(Who selects who is irrelevant. For simplicitity I am using "Male X selects", but it could just as easily be "Female X selects".)

1. Male Mage (50%) selects Female Mage (50%) - 25%
2. Male Mage (50%) selects Female Mundane with Mage Gene (25%) - 12.5%
3. Male Mage (50%) selects Female Mundane without Mage Gene (25%) - 12.5%
4. Male Mundane with Mage Gene (25%) selects Female Mage (50%) - 12.5%
5. Male Mundane with Mage Gene (25%) selects Female Mundane with Mage Gene (25%) - 6.25%
6. Male Mundane with Mage Gene (25%) selects Female Mundane W/O Mage Gene (25%) - 6.25%
7. Male Mundane W/O Mage Gene (25%) selects Female Mage (50%) - 12.5%
8. Male Mundane W/O Mage Gene (25%) selects Female Mundane with Mage Gene (25%) - 6.25%
9. Male Mundane W/O Mage Gene (25%) selects Female Mundane W/O Mage Gene (25%) - 6.25%

B - Results of Each Possible Pairing

*(This is going to assume 4 children from each possible pair. Additionally, this will use Punnett Square notation. M - Mundane gene; m - Mage gene)

• Male Mage 1 (mm) x Female Mage 1 (mm) - mm, mm, mm, mm (4 Mages)
• Male Mage 1 (mm) x Female Mage 2 (mm) - mm, mm, mm, mm (4 Mages)
• Male Mage 1 (mm) x Female Mundane W/ Gene (Mm) - Mm, Mm, mm, mm (2 Mages, 2 W/)
• Male Mage 1 (mm) x Female Mundane W/O Gene (MM) - Mm, Mm, Mm, Mm (4 W/)
• Male Mage 2 (mm) x Female Mage 1 (mm) - mm, mm, mm, mm (4 Mages)
• Male Mage 2 (mm) x Female Mage 2 (mm) - mm, mm, mm, mm (4 Mages)
• Male Mage 2 (mm) x Female Mundane W/ Gene (Mm) - Mm, Mm, mm, mm (2 Mages, 2 W/)
• Male Mage 2 (mm) x Female Mundane W/O Gene (MM) - Mm, Mm, Mm, Mm (4 W/)
• Male Mundane W/ Gene (Mm) x Female Mage 1 (mm) - Mm, Mm, mm, mm (2 Mages, 2 W/)
• Male Mundane W/ Gene (Mm) x Female Mage 2 (mm) - Mm, Mm, mm, mm (2 Mages, 2 W/)
• Male Mundane W/ Gene (Mm) x Female Mundane W/ Gene (Mm) - MM, Mm, mM, mm (1 Mage, 2 W/, 1 W/O)
• Male Mundane W/ Gene (Mm) x Female Mundane W/O Gene (MM) - MM, MM, Mm, Mm (2 W/, 2 W/O)
• Male Mundane W/O Gene (MM) x Female Mage 1 (mm) - Mm, Mm, Mm, Mm (4 W/)
• Male Mundane W/O Gene (MM) x Female Mage 2 (mm) - Mm, Mm, Mm, Mm (4 W/)
• Male Mundane W/O Gene (MM) x Female Mundane W/ Gene (Mm) - MM, MM, Mm, Mm (2 W/, 2 W/O)
• Male Mundane W/O Gene (MM) x Female Mundane W/O Gene (MM) - MM, MM, MM, MM (4 W/O)

Total Children born assuming 4 from every pairing: - 25 Mages - 30 Mundanes With the Gene - 9 Mundanes Without the Gene TOTAL: 64

Odds of a Mage being born - 39.0625%

Odds of a Mundane With the Gene being born - 46.8750%

Odds of a Mundane Without the Gene being born - 14.0625%

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B shows that assuming our sample (Generation 1) resulting in a 50/50 split between Mages and Mundanes AND assuming they all have 4 children with each possible mate, you will get a decrease in the percentage of Mages, in Generation 2. In Generation 3, we see that, doing the same process results in the same percentage and ratio (as well as birth defects from in-breeding [/AYO!]).

Of course, all of this is assuming specific circumstances. In reality, probability can't be assumed to go so clean-cut and clear. Environmental causes may very well cause Mages to swell or cull in numbers. You can't assume the statistic of what should happen in a controlled environment will always reflect what path reality takes.