Game with Six-Sided Die

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A game is played with a six-sided, regularly numbered die. The player starts with a number equal to 0.1n, where n is an integer between 1 and 6, inclusive. On each of 20 subsequent rolls, if the number rolled times 0.1 is greater than or equal to the player's current number, the player's current number is incremented by 0.1; if the number rolled times 0.1 is less than the player's current number and is odd, the player's number is decremented by 0.1; if the number rolled times 0.1 is less than the player's current number and is even, the player's number is unaffected. If 55% of the die rolls in a particular game are even, which of the following is a possible final value of that game?

I. 0.8

II. 0.5

III. 0.1

Review: Game with Six-Sided Die


Explanation

In this question, we are described a game that has a lot of instructions and doesn't seem very entertaining. The question poses the case that we roll 20 times and 55% of the rolls are positive, which means 11 of the 20 rolls are positive. We can imagine some cases: what happens if we roll 6 eleven times and 5 nine times? I'm looking for a maximum score here. If the player's score is 0.6 and he rolls a six, the score goes up to 0.7, but there's no case in which the score can advance further; it can either stay or go down. So Roman numeral I, a final score of 0.8, is impossible. That leaves only answer choice (D). There was a bit of luck here, but when analyzing by cases, extreme cases tend to be easier to test and tend to give insight into the situation. If we had to explore further we could have evaluated the game's minimum score.

The correct answer is (D).


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