It just means "probably". ruled that the businessman was probably involved ... , 'on the scales')" he found that the balance tipped toward the side which represented the businessman's involvement, not toward the side which represented the businessman's non-involvement.
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Would you write the words 'on the balace of probablity' instead of ' weighed the probabilites 'on the balance'?No. I would not write those words, nor 'weighed the probabilities ...".