AMC 10 · 2024 · #25
Grade 11 probabilitycountingPick an answer.
The phrase "the probability that two events are independent" sounds tangled, so the work is to make it concrete. Tool #12 (Draw a Venn Diagram) does that first: pattern and color are two overlapping properties, so the 6 balls sort into the four boxes striped-red, striped-blue, dotted-red, dotted-blue, and Tool #4 (Introduce a Variable) names those four counts a, b, c, d. Independence then becomes one equation in those four numbers. Tool #15 (Organize Information in More Ways) is the pivot and drives everything after it: the equation first appears as the lopsided 6a=(a+b)(a+c), but replacing the 6 by a+b+c+d collapses it to the symmetric ad=bc — equal diagonal products in the two-way table. That symmetric form is what makes the count short. Tool #7 (Identify Subproblems) then splits the count into two clean halves, tables with an empty row or column and tables with none, and Tool #2 (Make a Systematic List) finishes the second half, where only two shapes of positive four-part sums can reach 6 at all.
Sort the urn into four boxes
Balls fall into just four kinds.
Two yes-or-no properties always cut a group into exactly four pieces, so four numbers describe the urn completely.
10.S-CP.A.1Draw A Venn DiagramTurn independence into an equation
It becomes an equation in the counts.
With a uniform draw, every probability is a headcount divided by 6, so the whole probability statement is really a statement about integers.
9.A-CED.A.1Introduce A VariableRewrite as equal cross-products
It reduces to equal cross-products.
Swapping the anonymous 6 for a+b+c+d lets the clutter cancel and leaves a condition that treats all four boxes alike.
9.A-SSE.A.2Organize Information In More WaysSplit into flat tables and full tables
An empty box splits it into two families.
A product of two counts is zero only when a count is zero, so a single hole in the table always tears open an entire row or column.
A product of two counts is zero only when a count is zero, so a single hole tears open a whole row or column.
▸ Why?
Two nonzero numbers can never multiply to zero, so the zero has to come from one of the factors.
▸ Why?
The tables split into those with a hole and those without, and the two families never overlap.
Count the flat tables
After removing overlaps, 252.
Overlapping families get added once too often, and here every overlap is a single fully determined paint job.
10.S-CP.B.7Draw A Venn DiagramCount the full tables
Four shapes together give 720.
Once every box must be occupied, 6 is so small that only two shapes fit, and only one of them can balance the diagonals.
11.S-CP.B.9Make A Systematic ListAdd, reduce, read off m
Reducing leaves numerator 243.
A fraction over a power of 2 is in lowest terms as soon as the numerator is odd, and 3⁵ is odd.
11.S-CP.B.9Identify SubproblemsIndependence of "red" and "striped" is just the statement that the two diagonals of the striped-dotted by red-blue table have equal products, ad=bc, so count the tables that pass that test, weight each by how many of the 4096 flip outcomes build it, and 972/4096=243/1024 falls out.
- Sort the urn into four boxes
- Turn independence into an equation
- Rewrite 6a=(a+b)(a+c) as ad=bc
- Split into flat tables and full tables
- Count the flat tables: 252
- Count the full tables: 720
- Add, reduce, read off m