You peeked and saw the jackpot

Question

A host flips a fair coin that you do not see. On heads, the two envelopes hold $\$ 100 $and$ $200 $. On tails, they hold$ $200 $and$ $400 $. He hands you a random envelope, and you open it: it holds$ $400$.

Should you switch to the other envelope?

Accepted Answer

Use what the observed value tells you about the coin flip. The amount \400 appears only in the tails scenario, as the larger of the pair (\200, \400). It never appears under heads, where the amounts are \100 and \200. So seeing \400 means the coin landed tails and you are holding the larger envelope. The other envelope must hold \200. Switching would take you from \400 down to \200, a guaranteed loss. So you should not switch; keep the \400. This is the mirror image of the interior case. Had you instead opened \200, it could be the larger of the heads pair (other envelope \100) or the smaller…

You peeked and saw the jackpot

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