We are instructed to cover our thumbs during shmoneh esrei (Mishnah Berurah 95:6 quoting Ramak), havdalah (Mishna Brura 298:10) and in the beit hamikdash as these would select the kohen to do the avodah. What is the significance of covering the thumb and are there other times where one must do so?
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In the selection of kohanim in the mikdash they used a lottery. The "no thumbs" rule seems to serve the practical purpose of not allowing confusion or deception when fingers of many people are held out to be chosen. If a kohen were to put out a thumb it could be placed far enough away from his other finger (assuming many people's hands were tightly packed) to make it look like it belonged to another person, effectively giving him two entries in the lottery and upping his chances unfairly over his coworkers'.
Concerning covering the thumbs by havdala, the Magen Avraham 298:5 brings down the practice, sourcing the Zohar Vaykhel 208:1, and Pri Eitz Chaim shar shabbas chapter 24 Shar Hakavaanos regarding the procedure of what to do with ones hand during the bracha on fire at havdala.