Calculation for Determining a “Minority of a Minority”

The Beis Yosef (E”H 128) states that a “ten percent chance” is considered a “minority of a minority,” and even Rabbi Meir would not take that amount into consideration.

Sheorim Mitzuyanim B’halacha asks: Why should this be? According to his calculation, even a “fifteen percent chance” should be regarded as a “minority of a minority”? For sixty percent is considered a legitimate majority; this would result in “forty percent” being the minority. If we would divide the “forty percent-minority” into a majority and a minority, it would result in twenty-five percent joining the majority and fifteen percent becoming the “minority of a minority.” Why does the Beis Yosef state that ten percent only is a “minority of a minority,” when according to our calculation, even fifteen percent should be regarded as a “minority of a minority”?

Perhaps, it can be explained by saying that a recognizable majority is needed. Rabbeinu Yonah in Brochos states that a recognizable majority is when there is a two to one ratio. Accordingly, the original minority would be thirty-three percent. A minority of that would be a little more than ten percent. This could be the explanation for the Beis Yosef.