Dice Throws
A friend asked you to play a betting game with him. It involves rolling a fair 6-sided dice. You can choose 1 of the 3 options to try and win:
1 Throw to aim for an even number (Option 1)
2 Throws to aim for a square number (Option 2)
3 Throws to aim for a cube number (Option 3)
Which option should you choose to maximise your chances of winning?
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1 solution
Option 1 gives 1/2 chance, as you have 1 throw to get 2,4 or 6.
Option 2 gives 5/9 chance, because if you get 1 or 4 in your 1st throw, that's 1/3 chance, but if you fail and get 1 or 4 in your 2nd throw, that's 2/3 x 1/3 = 2/9. The sum of 1/3 and 2/9 is 5/9.
Option 3 gives 81/216, because if you get 1 in your 1st throw, that's 1/6 chance. If you get it in your 2nd throw, that's 1/6 x 5/6 = 5/36, if you get it in your 3rd throw, that's 1/6 x 5/6 x 5/6 = 25/216. The sum of these fractions is 81/216.
See which option gives a higher chance? That is none other than Option 2.