# Sunday Times Teaser 2774 – Loaded Dice

*by Ian Kay *

#### Published: 22 November 2015 (link)

I have two traditional-looking dice, but only one of them is fair. In the other there is an equal chance of getting a 1, 2, 3, 4 or 5, but the die is loaded so that a 6 is thrown more than half the time. I threw the two dice and noted the total. It turned out that with my dice the chance of getting that total was double what it would have been with two fair dice.

What (as a simple fraction) is the chance of getting a 6 with the loaded die?

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Here is a solution using Sympy:

I also coded a solution using SymPy, but it’s fairly easy to do the maths by hand:

I think it’s worth noting that if the unfair die were always to give a 6 (with a probability of 1, so 1 – 5 all have an equal probability of 0), then we get another solution where throwing a total of 10 is twice as likely with the unfair die as with both dies being fair.

My version is similar to Jim’s version