# Card increasing mystery

Intend there are 2 encounter down cards each with a favorable actual number and also with one two times the various other. Each card has value equivalent to its number. You are offered among the cards (with value $x$) and also after you have actually seen it, the supplier supplies you a possibility to exchange without any person having actually considered the various other card.

If you pick to exchange, your anticipated value needs to coincide, as you still have a $50\%$ opportunity of obtaining the greater card and also $50\%$ of obtaining the lower card.

Nonetheless, the various other card has a $50\%$ opportunity of being $0.5x$ and also a $50\%$ opportunity of being $2x$. If we maintain the card, our anticipated value is $x$, while if we exchange it, after that our anticipated value is: $$0.5(0.5x)+0.5(2x)=1.25x$$. so it feels like it is far better to exchange. Can any person clarify this noticeable opposition?

This mystery has constantly interested me. Something to think of is that there does not exist a consistent chance circulation over the favorable actual numbers (given that they are boundless). In getting to your mystery, it appears you are thinking that any kind of actual number is just as most likely, yet this can not hold true.

This problem is called both envelope mystery. This paper has a wonderful description of both envelope mystery, and also some referrals to more literary works pertaining to the problem.

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