Brainteasers & Puzzles

The Four Elements

Reveal shuffled cards until both water and earth appear, losing the moment fire shows. Wind is irrelevant, so the game reduces to whether fire comes last among the three live cards, and the winning probability is 1/3.

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Four cards are shuffled face down, showing water, earth, wind, and fire. You turn them over one at a time until you win or lose. You win once you have turned over both water and earth. You lose the moment you turn over fire. What is the probability that you win?

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#Only three cards matter

Wind decides nothing. Turning it neither wins nor loses, so strike it from consideration. You win exactly when both water and earth are revealed before fire, and you lose when fire comes up first. The result hangs entirely on the relative order of water, earth, and fire.

#The decisive order

You win if and only if fire is the last of those three to appear, since only then have both water and earth already shown. Among the three, each is equally likely to be last, so

P(win)=P(fire is last of the three)=13.(1)\PP(\text{win}) = \PP(\text{fire is last of the three}) = \tfrac13. \tag{1}
WEFwinWFEloseEWFwinEFWloseFWEloseFEWlose
Drop wind and list the six orders of water, earth, and fire. You win only when fire comes last, the two highlighted rows, so the probability is two in six, or one third.

#Only fire's position matters

The probability is 13\tfrac13. Wind is a red herring, and the whole game reduces to whether fire trails both water and earth, which happens one time in three.