We compute the probability that in 5 rolls of a fair six-sided die, exactly two distinct numbers appear.
["Title:\nComputing the Probability of Exactly Two Distinct Numbers in 5 Rolls of a Fair Die", "---", "Meta Description:\nLearn how to calculate the probability that exactly two distinct numbers appear when rolling a fair six-sided die five times. Discover the step-by-step combinatorial approach and formula.", "---", "Rolling a fair six-sided die five times is a classic probability problem with interesting combinatorial structure. A common question is: What is the probability that exactly two distinct numbers appear among the five rolls? Understanding this probability involves combining principles of combinatorics, counting favorable outcomes, and dividing by total possible outcomes.", "In this article, we break down the calculation clearly and rigorously so you can master this probability problem.", "---", "### What Does the Problem Mean?", "You roll a fair six-sided die (with faces numbered 1 through 6) five times. We want the probability that:", "- Exactly two different numbers appear in total.\n- No more, no fewer.\n- Each number can appear any number of times (≥1), as long as only two unique values are present.", "For example, outcomes like\n(1, 1, 1, 2, 2) count — exactly numbers 1 and 2 appear.\nBut outcomes like (1, 1, 1, 1, 3) do not — only two distinct numbers are 1 and 3, but this is allowed only if all rolls are 1 or 3 with both present. However, if rolls are 1,1,1,1,3, then only two distinct numbers — so this is counted. But if all five are 1, it has only one distinct number and is excluded.", "---", "### Total Number of Possible Outcomes", "Each die roll has 6 possible outcomes. Rolling the die 5 times gives:", "[\n6^5 = 7776\n]", "total equally likely outcomes.", "---", "### Step 1: Count Favorable Outcomes (Exactly Two Distinct Numbers in 5 Rolls)", "To count favorable outcomes, follow these substeps:", "#### Step 1: Choose the two distinct numbers\nWe first select which two numbers appear. From 6 possible die faces, choose 2:", "[\n\binom{6}{2} = 15\n]", "There are 15 ways to pick the set ( {a, b} ), where ( a <br/>\ne b ) and ( a, b \in {1,2,3,4,5,6} ).", "#### Step 2: Distribute the 5 rolls between the two numbers\nWe want all 5 rolls to be either (a) or (b), and both numbers must appear at least once.\nThis means we exclude the two cases where all 5 are (a) or all 5 are (b).", "The number of sequences using only (a) and (b) (including both missing) is (2^5 = 32). Subtract 2 invalid cases:", "[\n2^5 - 2 = 32 - 2 = 30\n]", "So, for each pair ( {a, b} ), there are 30 valid sequences where both numbers appear in the 5 rolls.", "#### Step 3: Combine choices\nMultiply the number of number pairs by valid distributions:", "[\n\ ext{Favorable outcomes} = \binom{6}{2} \ imes (2^5 - 2) = 15 \ imes 30 = 450\n]", "---", "### Can Any Other Combinations Work?", "Suppose three distinct numbers appear — even if probabilities differ, this case is excluded by our “exactly two” condition. Similarly, four or five distinct numbers clearly exceed two and are invalid.", "Thus, only exactly two distinct values count.", "---", "### Final Probability Calculation", "Now divide favorable outcomes by total outcomes:", "[\n\ ext{Probability} = \frac{450}{7776}\n]", "Simplify the fraction:", "[\n\frac{450}{7776} = \frac{25}{432} \quad \ ext{(after dividing numerator and denominator by 18)}\n]", "---", "### Final Answer", "[\n\boxed{\frac{25}{432} \approx 0.05787}\n]", "So, the probability that exactly two distinct numbers appear in five rolls of a fair six-sided die is (\dfrac{25}{432}), or about 5.79%.", "---", "### Summary", "- Total outcomes: (6^5 = 7776)\n- Choose 2 numbers: ( \binom{6}{2} = 15 )\n- For each pair, count sequences with both numbers appearing (excluding all same): (2^5 - 2 = 30)\n- Total favorable: (15 \ imes 30 = 450)\n- Probability: ( \frac{450}{7776} = \frac{25}{432} )", "Mastering this method helps solve related problems in combinatorics, probability, and discrete math. Whether analyzing dice rolls or other counting problems, understanding how to count favorable outcomes with constraints is essential.", "---", "Keywords: probability of exactly two distinct numbers in 5 die rolls, fair die probability, combinatorics, inclusion-exclusion in probability, die roll probability, two distinct outcomes, mathematical probability explanation."]
