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# Probability

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If a 6, 7, or 8 shows up on a roll of a pair of dice, you win two dollars. If anything else shows up, you lose one dollar. If you roll the dice 300 times, what is your expected amount of winnings?

Jun 30, 2022

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Note that there is a $$4 \over 9$$ chance to roll a 6,7, or 8.

This means that the remaining $${5 \over 9}$$th of the time, you lose a dollar.

The expected value is the average of the outcomes, and it is: $${{{(2 \times {4 \over 9})} + ({-1 \times {5 \over 9}})} \over 2 } = {3 \over 18}$$

So, your expected amount of winnings is $${3 \over 18 } \times 300 = \color{brown}\boxed{50}$$

Jun 30, 2022