An interesting probability question
A group of people was surveyed about whether the council should collect recycling once a week instead on once a fortnight. Two-thirds of the people said 'Yes' and one-third said 'No'. A year later the same group of people was surveyed again and this time one-quarter of them had changed their minds, resulting in a tie between the 'Yes' and 'No' votes. Of the people who originally voted 'Yes', what fraction changed their minds?
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1 solution
(2/3) - (1/2) = 1/6, so if 1/6 of everyone changed their vote from 'Yes' to 'No', then the votes would be equal.
But 1/4 of everyone changed their vote, more than 1/6. Therefore, an additional equal number of people who originally voted 'No' or 'Yes' would have to change their votes. This would not affect the overall result, as for every person changing from 'No' to 'Yes', there is another changing from 'Yes' to 'No'.
The number of extra people needed to change their vote is simply (1/4) - (1/6) = 1/12. But half of these are voting 'Yes' and half are voting 'No'. So the number of people voting 'Yes' = (1/12) / 2 = 1/24.
The total number of people changing their vote from 'Yes' to 'No' is then (1/6) + (1/24) = 5/24. Thus, the fraction that change their minds (5/24) of those who originally voted 'Yes' (2/3) = (5/24) / (2/3) = 5/16.