A carnival game allows the player a choice of simultaneously rolling two, four, six, eight, or ten fair dice. Each die has six faces numbered 1 through 6, respectively. After the player rolls the dice, the numbers that appear on the faces that land up are recorded. The player wins if the greatest number recorded is 1 or 2. How many dice should the player choose to roll to maximize the chance of winning?

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The probability of winning decreases as more dice is rolled.

To win with one die, we would need to roll a 1 or a 2.

$$ P(\text{Win}) = \frac{2}{6} =\frac{1}{3}$$

As you roll more dice, the overall probability decreases since each subsequent die needs to also be a 1 or a 2.

$$ P(\text{Win}) = \frac{1}{3}\cdot \frac{1}{3} \tag*{two dice} $$ $$ P(\text{Win}) = \left(\frac{1}{3}\right)^4 \tag*{four dice} $$