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Math / Statistics / Grade 8 / mt8u2

Count Overlapping Events Once

Use inclusive OR without counting two-star paths twice.

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A star first OR a star second includes both. Add the first-star and second-star counts, then subtract their two-star overlap once. Keep the same sample space for every count. With no returns, exclude same-physical-tile pairs before counting. If an overlap is empty, adding the two counts is enough. Reset the stated bag for each question.

Worked example

A worked bag has S1, S2, C1 and C2. Keep the first tile out. Among twelve allowed paths, six start with a star and six end with a star. Two paths have stars in both positions. At least one star therefore has 6 + 6 - 2 = 10 paths, or 10/12. Adding six and six alone would count the overlap twice.

Worked example only: S1, S2, C1, C2. Keep the first tile out; same-name pairs are impossible. 12 allowed paths. Star first: 6; second: 6. Two-star overlap: 2. 6 + 6 - 2 = 10 paths. Rows name the first tile, columns the second. Each cell names its star count.
Worked example only. The questions use their own bags and return rules.
Question 1 A NEW bag has 3 stars and 2 circles, each equally likely. Draw twice, keeping the first tile out. How many ordered paths start with a star?
Question 2 Use that same bag and keep-out rule. How many paths have a star in BOTH positions?
Question 3 Use that same bag. What is the chance of a star first OR a star second, including both?
Question 4 Someone adds 12 first-star paths and 12 second-star paths to claim 24/20. What went wrong?
Question 5 A different bag has 1 star and 3 circles. Keep the first tile out. What is the chance of a star first OR second?