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Alice and Bob are rolling a die, Alice waits for 666 in a row, Bob waits for 456 in a row. Who will wait longer on average?

Apparently the answer is Alice, but I don't understand why. Once you hit your first number, which has probability $\frac{1}{6}$ each roll, surely the probability of hitting your next number is also $\frac{1}{6}$, and if you miss it then you're back to where you started? Why would these probabilities differ?

veiph
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1 Answers1

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The probabilities differ because there are more outcomes that benefit Bob than Alice.

Say you roll a $3$. Then neither of them has gotten what they want.

But say that instead, you rolled a $6$. Both Alice and Bob are on the right tracks;

Now say you roll a $4$! Alice won't have what she wants, but Bob still has a chance!

The problem statement must mean that they are throwing the 3 dice at the same time and that Alice wants a triple $6$ while Bob wants a $4$, a $5$ and a $6$.

If you label the dice as $A, B, C$, Alice wants

$$A = B = C = 6$$

but Bob wants any of

$$\begin{cases}A = 4, B = 5, C = 6\\ A = 4, B = 6, C = 5\\ A = 5, B = 4, C = 6\\ A = 5, B = 6, C = 4\\ A = 6, B = 4, C = 5\\ A = 6, B = 5, C = 4\end{cases}$$

RGS
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  • This is another part of the question: For each of them, compute the probability of having the required combination during the first four rolls. (This can shed some light to the reasons why Bob has a better chance to win) and the description of the game specifically mentions in a row which sort of implies the ordering matters I think. If your interpretation is the correct one, the question is worded really poorly, but nothing else makes sense to me :/ – veiph Jan 20 '17 at 12:06
  • @ollieh I totally understand what you mean, and the wording is, indeed, very poor, but by "in a row" they may mean "3 consecutive rolls". Other than that, I cannot give sense to the question – RGS Jan 20 '17 at 13:17