Question

select the correct answer from the drop-down menu.
dustin and melanie are playing a game where they roll 2 standard 6 - sided number cubes and find the sum of their outcomes. each player gets a chance to guess the correct sum, and the player who guesses the correct sum wins the game.
dustin decides to guess a sum of 6. melanie decides to guess a sum of 7.
which player made the better decision?
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Explanation:

Step1: Find total outcomes

When rolling 2 six - sided dice, the total number of possible outcomes is \(n(S)=6\times6 = 36\) since each die has 6 faces.

Step2: Find outcomes for sum = 6

The pairs \((x,y)\) such that \(x + y=6\) are: \((1,5)\), \((2,4)\), \((3,3)\), \((4,2)\), \((5,1)\). So the number of outcomes for sum \(= 6\) is \(n(6)=5\). The probability of getting a sum of 6 is \(P(6)=\frac{n(6)}{n(S)}=\frac{5}{36}\approx0.139\).

Step3: Find outcomes for sum = 7

The pairs \((x,y)\) such that \(x + y = 7\) are: \((1,6)\), \((2,5)\), \((3,4)\), \((4,3)\), \((5,2)\), \((6,1)\). So the number of outcomes for sum \(=7\) is \(n(7) = 6\). The probability of getting a sum of 7 is \(P(7)=\frac{n(7)}{n(S)}=\frac{6}{36}=\frac{1}{6}\approx0.167\).

Step4: Compare probabilities

Since \(\frac{6}{36}>\frac{5}{36}\) (i.e., \(P(7)>P(6)\)), Melanie's guess of 7 has a higher probability of occurring than Dustin's guess of 6. So Melanie made the better decision.

Answer:

Melanie