In a game there are three teams. They are asked to throw dice, and for each team to win:
- A needs one 6 from 6 dice
- B needs two 6's from 12 dice
- C needs three 6's from 18 dice
What are the chances of A winning?
I thought it was $1-\left(\frac56\right)^2$, but my book gives the answer as $\frac23$.
Hint: What is the probability that for a particular die, the number appearing is not a $6$?
What is the probability that for all six dice, the number appearing is not a $6$? (It's the previous answer raised to the sixth power.)
What is the probability that for at least one of the six dice, the number appearing is a $6$? (It's one minus the previous answer.)