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Math / Statistics / Grade 8 / mt8u1

Count Opposite Outcomes

Find an event by counting all allowed paths except its opposite.

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Explore these events in Chance Lab

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An event and its complement split all allowed outcomes without overlap. For two draws, the complement of at least one star is zero stars. The complement of two stars is zero OR one star. Subtract the complement count from the total, keeping the same denominator. Counting paths gives probability only when those paths are equally likely.

Worked example

A worked bag has S1, S2 and C1. Return the first tile. All nine ordered paths are equally likely. Only C1 then C1 has zero stars. At least one star therefore has (9 - 1)/9 = 8/9. This is a model chance, not a promise of eight qualifying pairs in the next nine runs.

Worked example only: S1, S2, C1. Return the first tile. 9 equally likely paths. Zero stars: 1/9. At least one: (9 - 1)/9. Eight qualifying 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 with returns. What is the chance of ZERO stars?
Question 2 Use the same 3-star, 2-circle returned bag. What is the chance of AT LEAST ONE star?
Question 3 A different bag has 5 circles and no stars. Return each tile. What is the probability of at least one star in two draws? Enter 0 or 1.
Question 4 For two draws, which event is the complement of TWO stars?
Question 5 In a different experiment, some physical tiles are more likely than others. Is counting qualifying pairs out of all pairs enough to find the probability?