# Long Division

Parliament is $$\dfrac{1}{3}$$ Conservative members, who are incapable of changing their minds about anything, and $$\dfrac{2}{3}$$ Labour members who change their minds completely at random (with probability $$50\%$$) between successive votes on the same issue.

A randomly chosen member is observed to have voted twice in succession the same way. What is the probability that he will vote in the same way next time?

Note: Votes in the UK parliament are called divisions, hence the title

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