Two fair dice are rolled. What is the probability of rolling a sum of 7? Explain

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We have two fair dice, and we want to find the probability of rolling a sum of 7. A fair die means each face has an equal probability of one-sixth. To solve this, we'll use the probability formula: favorable outcomes divided by total outcomes. Let's systematically analyze all possible outcomes. Now let's examine the complete sample space. When we roll two dice, we need to consider all possible ordered pairs. The first die can show any value from 1 to 6, and independently, the second die can also show any value from 1 to 6. This gives us 6 times 6 equals 36 total equally likely outcomes. Each cell in this grid represents one possible outcome, written as an ordered pair where the first number is the result of die 1 and the second number is the result of die 2. Now let's identify the favorable outcomes - those that give us a sum of 7. We need to systematically check each cell in our grid and calculate the sum. Starting from the top: 1 plus 6 equals 7, so (1,6) is favorable. Next, 2 plus 5 equals 7, so (2,5) is favorable. Then 3 plus 4 equals 7, making (3,4) favorable. Similarly, (4,3) gives us 4 plus 3 equals 7. Then (5,2) gives us 5 plus 2 equals 7. Finally, (6,1) gives us 6 plus 1 equals 7. Counting these highlighted outcomes, we find exactly 6 favorable outcomes out of our 36 total possible outcomes. Now let's calculate the probability using our formula. The probability of rolling a sum of 7 equals the number of favorable outcomes divided by the total number of outcomes. Substituting our values, we get 6 divided by 36. We can simplify this fraction by dividing both numerator and denominator by 6, giving us 1 over 6. Converting to decimal form, this equals approximately 0.167, or about 16.67 percent. This pie chart visualizes the probability - the yellow section represents the 1 in 6 chance of rolling a sum of 7, while the gray section shows the 5 in 6 chance of rolling any other sum. Let's explore why sum of 7 has the highest probability by examining all possible sums. When we create a frequency distribution of all possible sums from 2 to 12, we see an interesting pattern. Sum of 2 can only occur one way: (1,1). Sum of 3 can occur two ways: (1,2) and (2,1). As we move toward 7, the number of ways increases: sum of 4 has 3 ways, sum of 5 has 4 ways, sum of 6 has 5 ways, and sum of 7 has the maximum of 6 ways. After 7, the pattern reverses symmetrically: sum of 8 has 5 ways, sum of 9 has 4 ways, and so on, until sum of 12 which has only 1 way. This symmetric distribution around 7 explains why it has the maximum probability of one-sixth.