Probability distribution of sum (by just counting outcome) The basic game of craps works as follows. You throw two dice. If the sum of the two faces showing up is 7 or 8 , you win and the game is over. If the sum is 2,3 , or 12 , you lose and the game is over. If the sum is anything else \( (4,5,6,9,10 \), or 11\( ) \), that value becomes your "point. "You then keep throwing the dice until the sum matches your point or equals 7 . If your point occurs first, you win and the game is over. If 7 occurs first, you lose and the game is over. The data has been collected in the Microsoft Excel Online file below. Open the spreadsheet and perform the required analysis to answer the question below. Do not round intermediate calculations. Round your final answer to four decimal places. What is the probability that you win the game?

To determine the probability of winning in craps, we start by counting the possible outcomes when rolling two dice. With two dice, there are 36 total combinations (6 faces on the first die multiplied by 6 faces on the second). Winning occurs under several conditions: rolling a 7 or 8 immediately, or rolling a point (4, 5, 6, 9, 10, or 11) followed by rolling that point again before rolling a 7. By analyzing the outcomes: - Winning immediately: - For a sum of 7: 6 combinations (1-6, 2-5, 3-4, etc.) - For a sum of 8: 5 combinations (2-6, 3-5, etc.) This gives us 11 win combinations. - Losing immediately: - For sums of 2, 3, or 12: 4 combinations (1-1, 1-2, 2-1, etc.) This gives us 4 loss combinations. - Remaining outcomes (the points): - Sums of 4 (3 combinations), 5 (4 combinations), 6 (5 combinations), 9 (4 combinations), 10 (3 combinations), and 11 (2 combinations) for a total of 21 combinations. Therefore, to find the winning probability, we need the calculation of rolling the point before a 7 occurs for the remaining outcomes. In short, for the winning probability: \[ P_{win} = \frac{P(7) + P(8) + P(point)}{P(total)} \] Where \(P(7)\) and \(P(8)\) are immediate wins, and \(P(point)\) depends on the point rolled and the probability of rolling above before a 7 occurs. Generally, the probability of winning the game of craps is calculated around 49.3% or 0.4930 when rounded to four decimal places.