Before us is Mishnah 1 of chapter 16 of Maseches Yevamos. The Mishnah deals with a woman who is left in doubt as to whether she requires yibum, and with the law of a woman whose mother in law traveled overseas.
"HaIshah shehalach ba'alah v'tzaratah lim'dinas hayam" - a woman whose husband and co-wife went overseas:
The case is one in which her husband and her co-wife - and not she herself - went overseas, "u'va'u v'amru lah: meis ba'alech" - and they came and told her: your husband has died. The report that reached her concerns the husband alone, and she has no information whatsoever about the co-wife. In such a case the Mishnah rules: "lo tinasei v'lo tisyabem, ad shetei'da shema me'uberes hi tzaratah" - she may neither marry nor undergo yibum until she knows whether her co-wife is pregnant.
The explanation is as follows: until now the husband had no children, and therefore either she or her co-wife became obligated in yibum. But it is possible that the co-wife became pregnant and is about to give birth, in which case the husband does have a child and the woman is exempt from yibum. Hence the two sides of the doubt:
She does not undergo yibum - since perhaps she is not obligated in yibum at all.
She does not marry anyone else - since perhaps she really is obligated in yibum.
Why is she not even permitted to undergo chalitzah?
The Gemara explains that she is not permitted even to undergo chalitzah, because it would look as though there is a yavam here and she is fit to perform chalitzah. The reason is that a chalutzah is forbidden to a kohen, and here there is a concern that after the act of chalitzah it will turn out that the chalitzah was unnecessary - for it is possible that her co-wife gave birth to a child, or that she was never pregnant at all. Once it becomes clear that the chalitzah was unnecessary, the woman returns to being permitted to a kohen and we permit her. But then someone who saw the chalitzah and did not hear the announcement that it was unnecessary might think that she is a full chalutzah and that a chalutzah is permitted to a kohen. For this reason we do not allow her to perform chalitzah.
"Halchah chamosah" - her mother in law went overseas:
The next case in the Mishnah: the woman's mother in law went overseas. Her husband died without children, and he had no brothers either, but his mother is still alive. The question is whether his mother gave birth to a son, in which case there is a yavam who obligates her in yibum. Regarding this the Mishnah says: "einah chosheshes" - she need not be concerned that perhaps she gave birth. But "yatz'ah m'lei'ah - chosheshes" - if at the time the mother in law left for overseas she was already pregnant, there is reason to be concerned that perhaps she gave birth to a son. It is worth noting precisely that in this case the birth of just any child is not enough; specifically a son is required, since only a son creates the reality of a yavam. The birth of a daughter creates no obligation of yibum.
"Rabbi Yehoshua omer: ein chosheshin" - Rabbi Yehoshua says: we are not concerned:
Rabbi Yehoshua disagrees and holds that even if she left pregnant we are not concerned, because there are two doubts here:
Perhaps she did not give birth at all, but rather the child was miscarried in some manner or was a stillbirth.
And even if she did give birth - perhaps she gave birth to a daughter.
These two doubts prevent us from being concerned that the mother in law, even though she left pregnant, actually produced a yavam who would create an obligation of yibum for this woman. Therefore she is permitted to marry anyone else, whomever she wishes.
In summary: in this Mishnah we learned two cases of doubt. In the case of a woman whose husband and co-wife went overseas and a report reached her of the husband's death alone - she may neither marry nor undergo yibum until it becomes clear whether her co-wife is pregnant, and she is not even permitted to perform chalitzah, out of concern for appearances regarding the prohibition of a chalutzah to a kohen. In the case of a woman whose mother in law went overseas - if she left when she was not pregnant, the woman need not be concerned, and if she left pregnant, she must be concerned that perhaps she gave birth to a son; and Rabbi Yehoshua holds that even in this case we are not concerned, because of a double doubt: perhaps she did not give birth at all, and perhaps she gave birth to a daughter.