Question

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a dice game involves rolling two dice. a player who rolls a 3, 4, 10, 11, or 12 wins 5 points. a player who rolls a 5, 6, 7, 8, or 9 loses 5 points.
if the expected value of playing this game is to lose \\(\frac{5}{9}\\) a player will \\(\boldsymbol{\downarrow}\\) points by rolling a 2.

Explanation:

Step1: Count total dice outcomes

When rolling two dice, total possible outcomes are $6 \times 6 = 36$.

Step2: Count winning/losing/rolling 2 outcomes

• Winning sums (3,4,10,11,12): Number of ways = 2+3+3+2+1 = 11
• Losing sums (5,6,7,8,9): Number of ways = 4+5+6+5+4 = 24
• Rolling a 2: Number of ways = 1

Step3: Define variable for 2's points

Let $x$ = points for rolling a 2. Expected value formula:
$$\text{Expected Value} = \frac{\text{Win Points} \times \text{Win Ways} + \text{Lose Points} \times \text{Lose Ways} + x \times \text{Roll 2 Ways}}{\text{Total Outcomes}}$$
Substitute known values: $\frac{5}{3} = \frac{5 \times 11 + (-5) \times 24 + x \times 1}{36}$

Step4: Solve for x

Multiply both sides by 36: $5 \times 12 = 55 - 120 + x$
Simplify: $60 = -65 + x$
Solve for $x$: $x = 60 + 65 = 125$

Answer:

125