AMC 10 · 2011 · #9
Grade 7 probabilityPick an answer.
The key reframe (Tool #16) is that “product greater than zero” just means “both numbers have the same sign,” which splits cleanly into two separate jobs (Tool #7): both numbers negative, or both numbers positive. Each job is a small independent-probability calculation, and the two jobs never overlap, so their chances add. Tool #1 (Draw a Diagram) backs this up: picturing the interval as a number line makes the negative part (length 20) and positive part (length 10) easy to measure against the whole length 30.
Measure the negative and positive parts
The interval splits into two shares by sign.
On a uniform interval, the chance of a region is just its length divided by the whole length.
7.SP.C.7Draw A DiagramChance both picks are negative
Both picks negative happens with chance 4/9.
For independent events, the chance of both happening is the product of their separate chances.
For two picks made without regard to each other, the chance of both happening is the two chances multiplied.
▸ Why?
When one draw tells you nothing about the other, their chances combine by multiplying.
▸ Why?
On an evenly spread interval each chance is just a length divided by the whole length.
Chance both picks are positive
Both positive happens with chance 1/9.
Same-sign multiplied gives positive, so the both-positive case counts too.
7.SP.C.8Identify SubproblemsAdd the two separate cases
Adding gives 5/9, choice (D).
Chances of separate, non-overlapping cases add together.
4.NF.B.3Identify SubproblemsA product is positive only when both numbers share a sign, so find each same-sign chance and add them up.
- Measure the negative and positive parts
- Chance both picks are negative
- Chance both picks are positive
- Add the two separate cases
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