The game Trivia Crack asks a question and gives four possible answers. Assume that I do not know the answer, so my attempt would be a pure guess at a 25% chance of success.

I have two optional tools at my disposal:

  • A Bomb removes two wrong answers, leaving only two possible answers This improves my odds to 50%.
  • A Double Guess lets me guess twice, and my first wrong guess is not counted. Assuming my first guess is wrong, this would reduce it to three possible answers, improving my odds to 33%.

The Bomb costs 6 coins. The Double Guess only costs 3 coins.

Over time, which is the better strategy, assuming a finite supply of coins? Is it more cost effective to improve from 25% to 50% odds at 6 coins, or from 25% to 33% odds at 3 coins?


1 Answer 1


The double guess:

\begin{align} \mathbb{P}(right \; answer|Bomb) &= 0.5 \\ \mathbb{P}(right \; answer|double \; guess) &= (1/4)*1 + (3/4)*(1/3) = 0.25 + 0.25 = 0.5 \end{align}

So you get the same odds for half as much money. The second derivation follows (denoting $R$ as 'right answer', $FS$ as 'first shot', $SS$ as 'second shot') because the first and the second shot are independent if we don't have any knowledge. The result is then that

\begin{align} \mathbb{P}(R|DG) &= \mathbb{P}(R|FS) + \mathbb{P}(R|SS \text{ & not } FS) \\ &= \mathbb{P}(R|FS) + \mathbb{P}(R|SS)\mathbb{P}(R|\text{ not } FS) \\ &= (1/4)*1 + (3/4)*(1/3) \end{align}

  • $\begingroup$ Can you explain in lay terms how the double guess is 50% instead of 33%? That's non-intuitive, and I don't understand your proof. If I get the first guess of a Double Guess wrong, can't we just pretend like that never happened and make the second guess the only thing we consider? $\endgroup$
    – Deane
    Jan 22, 2016 at 14:27
  • $\begingroup$ You get a 0.25 chance of getting the right result on the first shot. Given that the first shot failed (which happens with probability 1-0.25 = 0.75) you have a one third chance. I will add this to the answer :) $\endgroup$
    – Jeremias K
    Jan 22, 2016 at 14:35
  • $\begingroup$ Let me know if it's still not clear, I will try to elaborate further then $\endgroup$
    – Jeremias K
    Jan 22, 2016 at 15:42

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