AMC 10 · 2017 · #10
Grade 7 probabilityPick an answer.
Two numbers picked at random become a single point whose two coordinates are those numbers, so Tool #1 (Draw a Diagram) turns the whole question into one picture: a rectangle holding every possible outcome. Because the picks are uniform, every spot in the rectangle is equally likely, so a probability is just an area divided by the rectangle's area. Tool #7 (Identify Subproblems) splits that rectangle along the diagonal L=C into a small "Chloe wins" triangle whose area is easy to find. Tool #16 (Count the Complement) then finishes fast: find the chance Chloe wins and subtract it from 1.
Make every outcome a point in a box
Every outcome is a point in a box.
A uniform random pair is like a dart equally likely to hit any point of the rectangle, so chance becomes area.
A uniform random pair is a dart equally likely to hit any point of the rectangle, so chance becomes area.
▸ Why?
When every point is equally likely, the chance of a region is its share of the whole.
▸ Why?
The two numbers are chosen without regard to each other, so every pairing fills the rectangle evenly.
Cut the box along the line L = C
The tie line cuts off a triangle.
The line L = C is the fair boundary; everything below it is exactly the outcomes where Chloe's number came out bigger.
6.G.A.1Identify SubproblemsChloe's winning chance is one quarter
That triangle is exactly a quarter of the box.
Because the tall side is exactly double the wide side, Chloe's triangle is one of four equal pieces of the box.
7.SP.C.7Identify SubproblemsTake the complement to finish
The complement gives 3/4, choice (C).
With ties impossible, exactly one of the two wins, so their chances must add up to a full 1.
7.SP.C.5Change Focus Count The ComplementWhen two random numbers are compared, picture them as a point in a rectangle — then the probability is just the slice of area where your condition is true.
- Make every outcome a point in a box
- Cut the box along the line L = C
- Chloe's winning chance is one quarter
- Take the complement to finish
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