Write down a few simple positive fractions.
Now create a new fraction, such that its numerator equals the sum of all the numerators you wrote down and the denominator equals the sum of all the denominators.
Example: you wrote down 1/3, 2/7, 4/15 so the new fraction is 7/25.
Prove: the new fraction is smaller than the largest on the initial list
and bigger than the smallest.
Consider the fractions as part of a probability problem. Overall probabilities are a weighted average of conditional probabilities, which are isomorphic to the pure arithmetic view of the fractions.
In the example, treat the 1/3 as the probabilty of drawing a red ball from an urn containing 1 red ball out of a total of 3. The 2/7 represents 2 red balls out of an urn containing 7 balls altogether and the 4/15 represents a third urn having 4 of its 15 balls red.
Now throw all the balls into one big urn. The denominator of the new probability of drawing out a red ball will be the sum of the denominators; the numerator will be the sum of the numerators, that is, the total number of red balls.
The probability is now of course that (1+2+4)/(3+7+15) = 7/25.
But look at it another way: There is a 3/25 probability that the ball you draw out of the large urn is one that originally came from the urn with 3 balls, a 7/25 probability it came from the urn with 7 balls and a 15/25 probability it came from the urn with 15 balls.
Each of these probabilities is multiplied by the corresponding original probability from the first urn it was in, that is, the conditional probabilities.
The overall probability is now seen as the weighted average of the original probabilities. As such, it is between the highest and lowest going into that mean.
Posted by Charlie
on 2018-01-03 15:54:38