Mishna Berura Source Sheet 160:11

(4) If there was a doubt whether any work has been done with the water or not, a doubt whether the water contains the requisite quantity or not, a doubt whether it is unclean or clean, in these cases the doubt is considered to be clean because they have said in a case of doubt concerning hands as to whether they have become unclean or have conveyed uncleanness or have become clean, they are considered to be clean. Rabbi Yose says: in a case [of doubt as to] whether they have become clean they are considered to be unclean...*

(1) ספק נעשה בהן מלאכה – for work defiles the water for pouring, as is taught in the first chapter [of Tractate Yadayim, see Mishnah 3].

(2) טמאים – invalid for pouring [water for ritual cleansing of the hands].

(3) ספיקו טהור – even if there is with the water that he poured [for the ritual cleansing of the hands] so many of these doubts. And is the doubt of its being purified as the Rabbis and Rabbi Yossi argue in the concluding clause [of the Mishnah]. But the Halakha is not according to Rabbi Yossi.

The Sages taught in a baraita: With regard to the three times that the defilement of the wife is stated in the passage, namely: “If she is defiled” (Numbers 5:27), “and he warns his wife, and she is defiled” (Numbers 5:14), and “when a wife being under her husband goes astray and is defiled” (Numbers 5:29), why are all three necessary? One is to prohibit her to her husband, and one is to prohibit her to her paramour, and one is to prohibit her from partaking of teruma, even if she is the wife or daughter of a priest. This is the statement of Rabbi Akiva. Rabbi Yishmael said: It is unnecessary to derive from a verse that it would also be prohibited for this woman to marry a priest, as it can be derived a fortiori: If a divorced daughter of a priest, who is permitted to partake of teruma, is nevertheless forbidden to marry into the priesthood, then with regard to this sota, who is forbidden to partake of teruma, is it not logical that it is also prohibited for her to marry into the priesthood? The baraita continues by citing additional expositions involving the verse: “And she is defiled” (Numbers 5:14): What is the meaning when the verse states with regard to the cases in which a husband can compel his wife to drink the bitter water of a sota: “And he warns his wife, and she is defiled; or if the spirit of jealousy comes upon him, and he warns his wife, and she is not defiled” (Numbers 5:14)? If she is defiled, why does she need to drink? And if she is not defiled, why does he make her drink? The baraita answers: The verse tells you that it is discussing a case when there is uncertainty as to whether the woman was faithful to her husband, yet it is prohibited for her to engage in sexual intercourse with her husband until the matter is clarified. From here you can derive the halakha in a case of uncertainty with regard to whether the carcass of a creeping animal has imparted ritual impurity: Just as in the case of a sota, where the Torah does not consider unwitting adultery like intentional adultery, and rape is not treated like a willing transgression, because if a married woman committed adultery unwittingly or was raped she is not punished, yet still the Torah considers an uncertain case of adultery like a certain violation inasmuch as the woman is forbidden to her husband until the truth is clarified; so too, with regard to a creeping animal or other agents of ritual impurity, where the Torah does consider unwitting contact with impure items like intentional contact, as one contracts impurity whether or not his contact was intentional and an accident is treated like willing contact, is it not logical that the Torah must also consider an uncertain case of transmission of ritual impurity like a case of certain contact with an impure item? Accordingly, any cases of uncertain ritual impurity should be treated like certain impurity. The baraita continues: And since the case of sota is the source for the halakha with regard to uncertain ritual impurity, the details of the halakha are also derived from the case of sota. Therefore, from the place that you came from, i.e., from the source, it is derived that just as the prohibition with regard to a sota applies only when the uncertainty arises in the private domain, i.e., when she has secluded herself with the alleged paramour, so too, uncertain contact with the carcass of a creeping animal renders an item impure only if the contact was in the private domain. And furthermore, just as a sota is an entity that has awareness in order for her to be asked whether she actually committed adultery, so too, contact with a creeping animal renders an item impure only if it is an entity that has awareness in order for it to be asked, i.e., a person, or an item that may have contracted impurity in a place where a person was present and could have known.

The baraita concludes: And from here the Sages stated that if an entity that has awareness in order for it to be asked may have contracted impurity in the private domain, its uncertain impurity renders it impure; but if it may have contracted impurity in the public domain, its uncertain impurity leaves it pure. And with regard to an entity that lacks awareness in order for it to be asked, whether the uncertainty arose in the private domain or in the public domain, its uncertain impurity is deemed pure, as it is not similar to a sota.

(7) These are the cases of doubtful uncleanness that the sages declared to be clean...A doubt concerning the hands as to whether they have contracted uncleanness, have conveyed uncleanness or have attained cleanness, they are deemed clean.

(1) A non-permissible [food item] that was mixed with a permissible item that is not of its type, for example prohibited fat that was mixed with meat, a non-Jew should taste it, if [the non-Jew] says that it has no fat taste or if he says that it has taste but the taste is "pagum" (somehow not good), permitted. And this is that it is not for the ends of improving. And it must be that [the non-Jew] doesn't know that he is being depended upon. And if there is not there a non-Jew to taste it, we measure it against 60. And likewise, if it is a [mixture of two things which are] of the same type, because one cannot really upon taste, it is measured against 60. {REMA: It is not done now to trust a non-Jew, and we measure against 60.}

(2) If it is was mixed with something of the same type and it was spilled, in a case where one can not deal with it to measure it, if it is known that it was majority permissible, [it is] permissible. And if it is not known that it was majority permissible, [it is] forbidden. Rema: And in the matter of "being of the same type" we follow the name, if it is the same then this is "of the same type," but we do not follow the taste if it is the same or not. But if was mixed with something that is not of the same type and it spilled in a case where one can not deal with it to measure it, even it is known that it was majority permissible, [it is] forbidden. And if it was mixed with "of its kind" and with "not of its kind" and it spilled in a case where one can not deal with it to measure it, and it is known that it was majority "of its kind", one views the "not of its kind" as though it were not [present], and the rest, "of its kind" is greater and negates it.

(3) When do we rule this way? When it spilled. But a forbidden food that mixed with a permitted food, and it is before us, and it is impossible to deal with it to measure it, even though it's [a case] of rabbinic prohibitions, it is forbidden.

(1) ...And the Poskim write that the reason for this is that a doubt (ספק) that is dependent on a lack of knowledge can not really be considered a doubt (ספק), for this is merely the [faulty] knowledge of stupid people , which is not true of the first case that comes from a doubt (ספק) connected to spilling the substance.

(15) When there is a doubt with regard to the water [used to wash one's hands] - e.g., whether it had been used for other tasks or not, whether it contained the required quantity or not, whether it [the water itself] is ritually pure or not, and, similarly, when a person has a doubt whether he washed his hands or not, [in all these instances,] his hands are considered to be pure, because all instances where doubt arises concerning the ritual purity of hands, the hands are considered to be pure.

(11) "If there is doubt concerning the hands as to whether they have contracted uncleanness, have conveyed uncleanness or have attained cleanness, they are deemed clean." "Any doubt that arose in a public domain is deemed clean. "A condition of doubt concerning an ordinance of the scribes": [For instance, he is uncertain whether] he ate unclean food or drank unclean liquids, whether he immersed his head and the greater part of his body in drawn water, or whether there fell on his head and the greater part of his body three log of drawn water, such a condition of doubt is deemed clean. But if a condition of doubt arose concerning a father of uncleanness even though it was only rabbinical, it is deemed unclean.

(2) [If] he [the Judge] erred in weighing of opinions, e.g., [in] a case regarding which there is a dispute among Tannaim or Amoraim15Likewise in the case of a controversy among the Codifiers nowadays — ShaK. and the Halachah was not decided in accordance with [the views of] either of them explicitly and he acted in accordance with [the view of] one of them and he knew notthat the general practice had already spread throughout the world he is absolved from making compensation.

Rema:

One should not say 'I will render a decision in agreement with whomever I desire' pertaining to a matter wherein there is a difference of opinion [among the authorities]...

And if it is concerning ritual law and it is a matter involving a Biblical prohibition, one should adopt the restrictive view; and if it involves a Rabbinical matter, one should adopt the lenient view and [provided] only if the two dissenting [authorities] are equals; but one should not rely on the statements of a minor [authority] as opposed to the statements of one superior to him in wisdom and in numbers [of disciples or followers...

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Rabbi Abbahu said, and some say it was taught in a baraita: Three matters bring a person to a state of poverty as a divine punishment from Heaven: One who urinates before his bed while naked, and one who demeans the ritual washing of the hands, and one whose wife curses him in his presence.

With regard to one who demeans the ritual washing of the hands, Rava said: We only said this statement in a case where he does not wash his hands at all; however, if he washes his hands and does not wash them with a significant amount of water, we have no problem with it.

The Gemara notes: And that is not so, as Rav Ḥisda said: I wash my hands with handfuls of water and they gave me in reward handfuls of prosperity. Apparently, in order to garner the benefits of ritual washing of his hands, one should use a significant amount of water.