In a game, two standard six-sided dice are thrown. One is painted red Each die is numbered from 1 to 6 . The red die is thrown first followe Three people, Jan, Joe and Jim play the game. If the total is more than 6 , Jan wins. If one of the dice is a 2 or a 3 then Joe wins. If the number on the red die is greater or equal to the number on a) Using any suitable method, show who has the best chance

To determine who has the best chance of winning the game among Jan, Joe, and Jim, we need to analyze the outcomes of rolling two six-sided dice (one red and one standard die) and see how many outcomes favor each player. ### Total Outcomes When rolling two six-sided dice, there are a total of \(6 \times 6 = 36\) possible outcomes. ### Winning Conditions 1. **Jan wins** if the total of the two dice is more than 6. 2. **Joe wins** if at least one of the dice shows a 2 or a 3. 3. **Jim wins** if the number on the red die is greater than or equal to the number on the other die. ### Analyzing Each Player's Winning Conditions #### 1. Jan's Winning Outcomes Jan wins if the total is greater than 6. We can find the combinations that yield totals of 7 or more: - Total = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes - Total = 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes - Total = 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes - Total = 10: (4,6), (5,5), (6,4) → 3 outcomes - Total = 11: (5,6), (6,5) → 2 outcomes - Total = 12: (6,6) → 1 outcome Adding these up: - Total outcomes for Jan = \(6 + 5 + 4 + 3 + 2 + 1 = 21\) #### 2. Joe's Winning Outcomes Joe wins if at least one die shows a 2 or a 3. We can count the outcomes where at least one die is a 2 or a 3. - Outcomes with a 2: (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) → 6 outcomes - Outcomes with a 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) → 6 outcomes - Outcomes with a 2 on the second die: (1,2), (4,2), (5,2), (6,2) → 4 outcomes (excluding (2,2) already counted) - Outcomes with a 3 on the second die: (1,3), (4,3), (5,3), (6,3) → 4 outcomes (excluding (3,3) already counted) Total outcomes for Joe: - Total = \(6 + 6 + 4 + 4 = 20\) #### 3. Jim's Winning Outcomes Jim wins if the number on the red die is greater than or equal to the number on the other die. We can analyze this by considering each possible value of the red die: - Red die = 1: Jim wins 0 times (1 < 1) - Red die = 2: Jim wins 1 time (2 ≥ 1) - Red die = 3: Jim wins 2 times (3 ≥ 1, 3 ≥ 2) - Red die = 4: Jim wins 3 times (4 ≥ 1, 4 ≥ 2, 4 ≥ 3) - Red die = 5: Jim wins 4 times (5 ≥ 1, 5 ≥ 2, 5 ≥ 3, 5 ≥ 4) - Red die = 6: Jim wins 5 times (6 ≥ 1, 6 ≥ 2, 6 ≥ 3, 6 ≥ 4, 6 ≥ 5) Calculating the total: - Total outcomes for Jim = \(0 + 1 + 2 + 3 + 4 + 5 = 15\) ### Summary of Winning Outcomes - Jan: 21 outcomes - Joe: 20 outcomes - Jim: 15 outcomes ### Conclusion Jan has the best chance of winning with 21 favorable outcomes, followed by Joe with 20 outcomes, and Jim with 15 outcomes. Therefore, **Jan has the best chance of winning the game**.