Gemara

(Mishnah): If one buys a cow or female donkey less than three years old (from a Nochri), its first child (if it is a male) is a Vadai Bechor (a firstborn; it is automatically a Korban), so he must give it to a Kohen. If it is above three years, its first child is a Safek (he keeps it, and eats it after it gets a Mum (blemish)).

Bechoros 19b (Mishnah - R. Yishmael): If one bought an animal from a Nochri and is unsure whether it already gave birth, if it was a goat (and it gave birth) in its first year, the kid is given to a Kohen. Past one year, it is a Safek. For a sheep, within two years the lamb is given to a Kohen. After this, it is a Safek. For a cow, within three years, the calf is given to a Kohen. After this, it is a Safek;

R. Akiva: Not only a proper child exempts from Bechorah! Even Tinuf (a dissolved fetus; alternatively, bubbles of blood) in a small animal is a sign of a (miscarried) child, and a Shilya (fetal sac) in a large animal is a sign of a child;

The rule is, if we know that an animal already gave birth (or miscarried), the Kohen gets nothing. If we know that it did not give birth before, the Bechor belongs to the Kohen. If we are unsure, the owner eats it when it gets a Mum.

Question: Why does R. Yishmael say that after one year, it is a Safek? Most goats become pregnant and give birth in their first year!

Answer #1: He holds like R. Meir, who is concerned for the minority.

Answer #2 (Ravina): He can hold like Chachamim. Chachamim follow a majority that does not depend on an act, but not a majority that depends on an action (an animal must breed to become pregnant and bear a child).

20a (Beraisa - R. Yehoshua): If a six month goat had Tinuf, it can give birth in its first year. If a yearling sheep had Tinuf, it can give birth in its second year.

R. Akiva says, I argue with this. Rather, if we know that an animal previously gave birth, the Kohen does not receive anything...

20b (R. Chanina of Sura): They argue about whether milk exempts from Bechorah. R. Akiva alludes to another way of knowing that an animal already gave birth, i.e. milk. Most animals do not have milk before giving birth. R. Yehoshua is concerned for the minority of animals that have milk before giving birth.

Question: R. Yehoshua is not concerned for the minority!

Yevamos 119a (Mishnah): If a (childless) man went overseas with a wife, and left a wife Leah home, and he died, Leah may not remarry nor do Yibum until she hears whether or not the other wife is pregnant.

If he did not have any brothers and his mother was overseas, Leah need not be concerned lest his mother gave birth to a boy before he died. If his mother was pregnant, Leah must be concerned.

R. Yehoshua says she need not be concerned (for the minority. Half of all babies born are females, and a minority of pregnant women miscarry, so a minority give birth to males).

Answer: The opinions in our Beraisa must be switched.

Support (Beraisa - R. Yehoshua): Milk exempts from Bechorah;

R. Akiva says, it does not exempt.

Rishonim

Rambam (Hilchos Bechoros 4:7): If one buys an animal from a Nochri and does not know whether or not it already gave birth, if it had a (male) firstborn it is a Safek. The owner eats it after it gets a blemish. It does not belong to a Kohen, for ha'Motzi mi'Chavero Alav ha'Re'ayah.

Rosh (Bechoros 3:1): Even after a year (for a goat, or two years for a lamb...), it is a Safek. The Halachah follows R. Yishmael regarding this, for we do not find that anyone disagrees. Within a year, the Halachah follows R. Yehoshua and R. Akiva, who are concerned for TInuf, for they are a majority against R. Yishmael. R. Yonah says that even though usually we hold like Chachamim who follow the majority, Tinuf is not a minority, for it is common. This is difficult. The Gemara calls it a minority! Rather, Rashi explains that the majority of animals become pregnant in the proper time, and this supports the minority of Tinuf. We must say so to resolve R. Yehoshua, for regarding a Yevamah he is not concerned for the minority. One who buys an animal from a Nochri eats the Bechor after it gets a Mum. Even the one who is not concerned for Tinuf agrees that the Kohen does not get it, for we do not follow the majority in monetary cases.

Rosh (2 and Yevamos 16:1): R. Tam rules that milk does not exempt, for below we rule like R. Shimon ben Gamliel, who exempts a nursing mother because it has mercy only on its own child. Milk did not suffice to exempt! He infers that we are concerned for a minority when a Chazakah (that it did not give birth) supports it. This is difficult. If so, what forced the Gemara to switch the opinions because R. Yehoshua is not concerned for the minority regarding a Yevamah? There, there is no Chazakah! In the Reisha there, he is concerned for the minority who miscarry (and do not bear children), because the Chazakah that Leah will fall to Yibum supports it! The Gemara says that the Reisha is like R. Meir, who is concerned for the minority. Rather, in Bechoros, the mother's Chazakah does not support the minority, for there is an opposing Chazakah that the child is Chulin, like it was in the womb. The latter Chazakah is stronger, for Chezkas Mamon supports it (it belongs to the owner, and he need not give it to a Kohen. However, elsewhere R. Shimon ben Gamliel is not concerned for the minority! Rather, R. Yochanan rules like R. Shimon ben Gamliel's reasoning that an animal would not nurse another's baby for a case that we know that the mother had milk before it gave birth. If later it nurses, we know that it gave birth. He agrees that milk exempts. The Ramban rules like this.

Poskim

Shulchan Aruch (YD 316:1): If one buys an animal from a Nochri and does not know whether or not it already gave birth, if it had a child, it is a Safek Bechor. The owner eats it after it gets a blemish. It does not belong to a Kohen, for ha'Motzi mi'Chavero Alav ha'Re'ayah. This is whether he bought it within the first year or afterwards.

Beis Yosef (DH ha'Loke'ach): The Gemara connotes that the Halachah follows R. Akiva. The Rambam and Rosh rule like him.

Shulchan Aruch (3): An animal with milk is exempt, for most animals do not have milk until they give birth.

Rema: Some say that we do not rely on milk. The Acharonim rule like this, and this is the custom in these lands.

Shach (3): We do not rely on milk even if the animal is very old.

Gra (5): Even though we do not hold like R. Meir who is concerned for the minority, we are concerned when a Chazakah supports it. This is only to be stringent, but not to be lenient. If a child is near a dough, we do not join Chezkas Taharah of the dough to the minority of children who do not touch (Kidushin 80a). An uncommon minority does not join even to be stringent. This is why the Chezkas Isur of a live animal does not join with the tiny minority of amateurs who slaughter. (Rather, we assume that an expert slaughtered.) We are concerned for a tiny minority only regarding Eshes Ish, e.g. if a man sank in water and we cannot see the end, or concern that one gave gifts to his bride before Kidushin. Tosfos and the Rosh say that the Sugya in Yevamos is like R. Meir. The Ramban, Rashba and Nimukei Yosef say that it is even like Chachamim; they agree in such a case. All agree that we are concerned for a common minority even without a Chazakah. This is why we must check the lungs, even when there is a Chazakah against it. (This requires investigation.)

Note: Perhaps the Chazakah is for an animal that gave birth. It is Muchzak from then to be Kosher. Alternatively, a slaughtered animal is Muchzak to be Kosher, until we find that it is Tereifah (Chulin 9a).

Rema: However, if there is another reason to permit, or the Nochri speaks Lefi Tumo (unaware of the consequences) without intent to make it worth more, we rely on this to permit. This refers to cows, in places where we did not hear that they have milk before giving birth. Goats often have milk before giving birth, so we do not rely on this, even if it is a Sefek Sefekah, e.g. it gave birth to two (and we do not know which was born first).

Pischei Teshuvah (2): Tzemach Tzedek (64) derives from Terumas ha'Deshen (167) that also the other reason for Heter must suggest that it is not a firstborn. A Safek that a Nochri owns the mother does not permit. R. Akiva Eiger (37) and Levushei Serad (113) disagree, for in any case it is a Sefek Sefekah.

Shach (7): He does not discuss a proper Sefek Sefekah, for this always permits! Rather, he refers to a Sefek Sefekah like this case. The Terumas ha'Deshen says that the first Safek is mid'Oraisa and it is not Batel even in a majority, and all the more so if it is mixed with only one other calf.