\n\n$$\\text{Probability} = \\frac{\\text{Successful Outcomes}}{\\text{Total Outcomes}} = \\frac{3}{9}$$\n\nWe can simplify this fraction by dividing both the numerator and the denominator by $3$:\n\n$$\\frac{3}{9} = \\frac{1}{3}$$\n\nEither $\\frac{3}{9}$ or $\\frac{1}{3}$ is accepted for full marks."}]},{"id":"c203c9f3-cf48-4333-b151-6a9aeec6b1a7:caseanalysis","label":"(b) Case-by-Case Analysis","steps":[{"type":"text","order":0,"title":"Find total possible outcomes","content":"To begin, let's find the total number of possible combinations when picking one tile from each set. \n\nSince Set 1 contains $3$ tiles ($\\{4, 7, 10\\}$) and Set 2 contains $3$ tiles ($\\{1, 5, 8\\}$), the total number of possible outcomes is:\n\n$$\\text{Total outcomes} = 3 \\times 3 = 9$$","highlights":[{"text":"Set 1\nContains tiles with numbers 4, 7, and 10.","location":"specification"},{"text":"Set 2\nContains tiles with numbers 1, 5, and 8.","location":"specification"}]},{"type":"text","order":1,"title":"Analyze Case 1: Tile 4","content":"Let's check each number in Set 1 case-by-case to see which partners from Set 2 make a sum of **less than 10**.\n\n**If Leo chooses the tile with $4$ from Set 1:**\nWe need a partner $x$ from Set 2 such that $4 + x < 10$.\n\n* With $1$: $4 + 1 = 5$ (which is $< 10$) $\\rightarrow$ **Valid**\n* With $5$: $4 + 5 = 9$ (which is $< 10$) $\\rightarrow$ **Valid**\n* With $8$: $4 + 8 = 12$ (which is $\\ge 10$) $\\rightarrow$ Invalid\n\nThere are **$2$** successful outcomes for this case: $(4, 1)$ and $(4, 5)$."},{"type":"text","order":2,"title":"Analyze Case 2: Tile 7","content":"**If Leo chooses the tile with $7$ from Set 1:**\nWe need a partner $x$ from Set 2 such that $7 + x < 10$.\n\n* With $1$: $7 + 1 = 8$ (which is $< 10$) $\\rightarrow$ **Valid**\n* With $5$: $7 + 5 = 12$ (which is $\\ge 10$) $\\rightarrow$ Invalid\n* With $8$: $7 + 8 = 15$ (which is $\\ge 10$) $\\rightarrow$ Invalid\n\nThere is **$1$** successful outcome for this case: $(7, 1)$."},{"type":"text","order":3,"title":"Analyze Case 3: Tile 10","content":"**If Leo chooses the tile with $10$ from Set 1:**\nWe need a partner $x$ from Set 2 such that $10 + x < 10$.\n\nSince all numbers in Set 2 are positive, adding any of them to $10$ will result in a sum greater than $10$.\n\nThere are **$0$** successful outcomes for this case."},{"type":"text","order":4,"title":"Calculate the final probability","content":"Now we add up the favorable outcomes from all cases:\n\n$$\\text{Total favorable outcomes} = 2 + 1 + 0 = 3$$\n\nTo find the probability, we divide the number of favorable outcomes by the total possible outcomes:\n\n$$\\text{Probability} = \\frac{\\text{Favorable outcomes}}{\\text{Total outcomes}} = \\frac{3}{9}$$\n\nSimplifying the fraction gives:\n\n$$\\text{Probability} = \\frac{1}{3}$$"}]}],"generatedAt":"2026-06-13T01:55:49.248Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"c133f62e-6c2c-4039-b5ab-d483b1dd3dbc","revisionId":"c7ce8212-a3c5-4884-b6c2-44f5bd42a719","questionNumber":null,"questionType":"LA","specification":"Four cards are labeled with the numbers 3, 5, 7, and 9.","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"a9142079-7b04-41ea-b850-de1c86c70402","label":"a","content":"Write down all the possible **three-digit numbers** that can be created using these cards.","markscheme":"- 357, 359, 375, 379, 395, 397\n- 537, 539, 573, 579, 593, 597\n- 735, 739, 753, 759, 793, 795\n- 935, 937, 953, 957, 973, 975\n\nAward 1 mark for at least 6 correct numbers. M1\nAward 2 marks for at least 18 correct numbers. M1\nAward 3 marks for all 24 correct numbers. A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"At least 6 correct numbers listed.","code":"M"},{"id":"p2","marks":1,"description":"At least 18 correct numbers listed.","code":"M"},{"id":"p3","marks":1,"description":"All 24 correct numbers listed.","code":"A"}],

Systematic Listing

Question 22

Systematic Listing Questions

Systematic Listing

Question 22

Systematic Listing Questions