Question:** A fair 6-sided die is rolled 4 times. What is the probability that the sum of the rolls is exactly 14?
["Understanding the Probability of Rolling a Sum of Exactly 14 with a Fair 6-Sided Die (4 Rolls)", "When rolling a fair 6-sided die four times, many players wonder: What is the probability that the total sum equals exactly 14? This frequently asked question taps into fundamental concepts of probability, combinatorics, and integer partitioning under constraints. In this article, we’ll explore how to calculate this probability using clear mathematical reasoning.", "---", "### What Is the Sample Space?", "Rolling a 6-sided die four times means each roll results in an integer from 1 to 6. Since each roll is independent, the total number of possible outcomes is:", "$$\n6^4 = 1296\n$$", "We want to count how many of these outcomes produce a total sum equal to 14.", "---", "### Target: Sum of 14 with Four Rolls (Each Roll between 1 and 6)", "Let the outcomes be represented as four integers ( a, b, c, d ) such that:", "$$\na + b + c + d = 14 \quad \ ext{where } 1 \leq a, b, c, d \leq 6\n$$", "To simplify, make a change of variables: let ( x_i = a_i - 1 ). Then ( x_i \geq 0 ), and since ( a_i \leq 6 ), we have ( x_i \leq 5 ). The equation becomes:", "$$\n(x_1 + 1) + (x_2 + 1) + (x_3 + 1) + (x_4 + 1) = 14 \implies x_1 + x_2 + x_3 + x_4 = 10\n$$", "Now, we want the number of integer solutions to:", "$$\nx_1 + x_2 + x_3 + x_4 = 10 \quad \ ext{with } 0 \leq x_i \leq 5\n$$", "---", "### Step 1: Count Total Non-Negative Integer Solutions Without Upper Bounds", "First, compute the total number of non-negative integer solutions to ( x_1 + x_2 + x_3 + x_4 = 10 ) with no upper limits. This is a classic “stars and bars” problem:", "$$\n\ ext{Total} = \binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} = 286\n$$", "---", "### Step 2: Subtract Invalid Cases (Where One or More ( x_i > 5 ))", "We now subtract cases where one or more variables exceed 5 (i.e., ( x_i \geq 6 )), since ( x_i \leq 5 ) is required.", "Let’s use the principle of inclusion-exclusion.", "(a) One variable ≥ 6:\nSuppose ( x_1 \geq 6 ). Let ( x_1' = x_1 - 6 ), so ( x_1' \geq 0 ). The equation becomes:", "$$\nx_1' + x_2 + x_3 + x_4 = 4\n\Rightarrow \binom{4 + 4 - 1}{3} = \binom{7}{3} = 35\n$$", "There are 4 variables, so total such cases: ( 4 \ imes 35 = 140 )", "(b) Two variables ≥ 6:\nSuppose ( x_1 \geq 6, x_2 \geq 6 ). Let ( x_1' = x_1 - 6, x_2' = x_2 - 6 ), so equation:", "$$\nx_1' + x_2' + x_3 + x_4 = 10 - 12 = -2\n$$", "No solutions (negative sum), so contributions from this and higher overlap are zero.", "Thus, by inclusion-exclusion, number of invalid solutions is exactly 140.", "---", "### Step 3: Valid Solutions", "Valid solutions = Total unrestricted – Invalid:", "$$\n286 - 140 = 146\n$$", "So, there are 146 favorable outcomes where four die rolls sum to 14.", "---", "### Step 4: Compute Probability", "The probability is the ratio of favorable outcomes to total outcomes:", "$$\nP(\ ext{sum} = 14) = \frac{146}{1296}\n$$", "Simplify the fraction:", "Divide numerator and denominator by 2:", "$$\n\frac{73}{648}\n$$", "This is the exact probability.", "---", "### Final Answer:", "The probability that the sum of four rolls of a fair 6-sided die is exactly 14 is:", "$$\n\boxed{\frac{73}{648}} \approx 0.1126 \ ext{ or } 11.26%\n$$", "---", "### Why This Matters", "Understanding such probabilities enhances strategic decision-making in games involving dice, improves statistical intuition, and strengthens skills in combinatorial reasoning. Whether you're a banjo enthusiast calculating odds while playing, or a student mastering chance and randomness, mastering these calculations empowers better analysis of uncertainty.", "---", "Keywords: probability of 14 with four 6-sided die rolls, fair die probability, combinatorics sum 14, die roll sum calculation, probability example, total outcomes 6^4, valid dice combinations, integer partition sum constraint", "Meta Title: Probability sum 14 with four dice – exact calculation & explanation\nMeta Description: Learn how to calculate the probability of rolling a sum of 14 with four fair 6-sided die rolls using combinatorics and inclusion-exclusion. Exact solution with steps."]
