What is wrong with this?

Discrete Mathematics Level 3

Alexander wants to find the probability that when 10 people were to each roll a regular 6-sided die, the first player has the highest value. He thinks that the answer is \( \frac1{512} \), and has come up with the following solution:

Step 1. This is equivalent to the first player out-rolling the second player, the first player out-rolling the third player, and so on.

Step 2. The probability that the first player rolls a higher number than the second does is \(\frac{1}{2}\), and same for the other players.

Step 3. Thus, the overall probability is \(\left(\frac{1}{2}\right)^9={\frac{1}{512}}\).

Which of these 3 steps is wrong?

Note: When evaluating a step, you should assume that the other steps are valid.


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