Two symmetric cubical dice are rolled once. Match the items of Column-I with the items of Column-II.
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2|c|Column-I & 2c|Column-II
A & Probability that the numbers appearing are equal & I & 1/12
B & Probability that the numbers are all distinct & II & 5/36
C & Probability that the sum of numbers is 10 & III & 1/6
D & Probability that the sum of numbers is 6 & IV & 4/36
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Show Hint
For sum-based problems with two dice, the number of outcomes for a sum $S$ follows a symmetric pattern: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 for sums 2 through 12, respectively.
Step 1: Count the sample space. Two distinct dice give $6\times6=36$ equally likely ordered outcomes. Step 2: Event A, equal numbers. The doubles are $(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)$, so $6$ outcomes give $P=\tfrac{6}{36}=\tfrac16$, matching III. Step 3: Event B, all distinct. Distinct means not equal, the complement of A, so $P=1-\tfrac16=\tfrac56$, matching V. Step 4: Event C, sum is 10. The pairs are $(4,6),(5,5),(6,4)$, that is $3$ outcomes, so $P=\tfrac{3}{36}=\tfrac1{12}$, matching I. Step 5: Event D, sum is 6. The pairs are $(1,5),(2,4),(3,3),(4,2),(5,1)$, that is $5$ outcomes, so $P=\tfrac{5}{36}$, matching II. Step 6: Assemble the matching. We get A-III, B-V, C-I, D-II, which corresponds to option 2. \[ \boxed{\text{A-III, B-V, C-I, D-II}} \]
