dice biased such that a 3 comes up twice as often as each other number?

Question

I just attempted to solve a problem that involved the title.

So, it turns out I was getting an incorrect answer b/c I computed the probability of getting a 3 as 2/6 as opposed to 2/7. I don't fully understand why the probability of getting a 3 would be 2/7 though.

If I think about it like if you had the numbers 1-6 in a bag and you want a 3 to come up twice as often, the probability would be like adding another 3 to the bag. Giving it a probability of 2/7 where as if you said the probability was 2/6 it would be like pulling out one number and substituting a 3 for it. This makes sense too me but, in the case of a dye you can't really add another side even though it is biased. So why isn't it 2/6? What does it mean for a dice to have a 2/6 probability of landing on one side as opposed to 2/7?

Answer 1

Okay so you understood probability good, but to some extend you are asking a wrong question. The problem is, that when we say a dice which comes up 3 twice more often than any other side, you imagine a standard 6 sided die. So the question you must be asking is in fact, how can at all there exists a dice that lands twice more on 3 than other side, if it is physically a dice (which is what you are imagining).

Whenever you do probabilities, try to forget about the real world, and work only in the abstract world where they tell you to. In some sense you can view the dice, just as a some blackbox, which instead of being rolled physically, jsut spits out numbers. Then if you think about it if the probability of the numbers $1,2,4,5,6$ is $p$ and then of $3$ is $2p$ it is required that $p+p+2p+p+p+p=1$ so when you solve you'll get $p=1/7$ and so probability for $3$ is $2/7$. I'm not sure whether that made it more clear, but I hope so.

Answer 2

Hint:

Dice number | Relative Probability
------------+---------------------
     1      |        1
     2      |        1
     3      |        2
     4      |        1
     5      |        1
     6      |        1
------------+---------------------
   Total    |        7