import math

def rankByPDF(alpha, beta):
    mean = alpha / (alpha + beta)
    factor = 1.65 * math.sqrt( (alpha * beta) / ( ( (alpha + beta)**2)  * (alpha + beta + 1) ) )
    return mean - factor

##
## Property A: 20 5-Stars and 2 1-Stars
## Alpha = 1 + (20 * 1) + (2 * 0.2) = 21.4
## Beta = Alpha - 22 (count of all reviews) = 1.6
##
print "Property A: " + str( rankByPDF(21.4, 1.6) )

##
## Property B: 6 5-Stars and 1 3-Stars
## Alpha = 1 + (6 * 1) + (1 * 0.6) = 7.2
## Beta = Alpha - 7 (count of all reviews) = 0.8
##
print "Property B: " + str( rankByPDF(7.2, 0.8) )

## Output
## Property A: 0.844747303679
## Property B: 0.735