A paradox of changing odds I read about - doing my head in, but it must be easy to explain why.
Three sets of two playing cards: AA KK AK
With the cards turned face down, you task is to pick the AK pair. Odds are 3 to 1 you pick the correct pair. That is not the problem.
You pick a pair, and one card is turned over - it's a K. That means now, the odds of having the AK pair are 2 to 1.
How? Nothing changed, no magic, yet just by seeing one card of the pair you chose the odds change from 3 to 1 -> 2 to 1.
I have read the solution, but still don't understand this simple logic.
Nick
Answer
Now you know that the pair is not AA. That new information could change the odds. Usually the original odds would be quoted as 2 to 1 against. In fact if you turn just one card at random and find a K, the odds you have AK are still 2 to 1 against. You now have $2/3$ chance of having the KK and $1/3$ chance of having AK because it is twice as likely you found a K from KK. To see this in detail, list the six possibilities of which pair you pick and which card you pick from the pair. Initially two of the six have you picking AK. When you find a K, three of the possibilities are ruled out, but only one of the remaining three has you choosing AK.
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