AMC 10 · 2015 · #15
Grade 7 probabilityPick an answer.
Work backwards from the GPA target first: two of the four grades are already settled, so the condition on the GPA collapses into a single condition on the English and History points. Each of those classes has exactly three possible grades, so the whole sample space is only nine pairs — small enough to write out completely, which is the safest way to be certain about the boundary. The boundary is the entire problem. An A paired with a C scores exactly 6 and passes, while a B paired with a C scores 5 and fails, so the intuitive reading 'she needs to do well in both' is wrong. Before any of that, the unstated C probabilities have to be recovered as the leftovers from 1, which is a subproblem in its own right. Independence turns each cell into a product, and computing the three failing cells as well as the six passing ones gives a check that costs almost nothing.
Turn the GPA target into a point target
The average becomes a plain point target.
Two of the four grades are already decided, so they are just a fixed head start that can be subtracted straight off the target.
7.EE.B.4Work BackwardsRecover the missing C probabilities
Each missing probability is recovered.
When the list of possible outcomes is complete, the one probability you were not told is forced by the others.
7.SP.C.7Identify SubproblemsList all nine grade pairs
There are only nine grade pairs.
Nine outcomes is few enough to write out in full, which removes every doubt about which side of the cutoff a pair lands on.
7.SP.C.8Make A Systematic ListGive each passing pair its probability
Independence prices each pair directly.
Independent means one grade tells you nothing about the other, so the two chances simply multiply.
Independent means one grade tells you nothing about the other, so the two chances simply multiply.
▸ Why?
When one outcome carries no information about the other, the chance of both is the product of the two.
▸ Why?
Every grade in one subject can pair with every grade in the other, so the outcomes fill a full grid.
Add the passing cells
Adding the passing cells gives 11/24.
One shared denominator turns six fraction additions into a single addition of whole numbers.
5.NF.A.1Organize Information In More WaysCross-check with the failing cells
The failing cells confirm it, choice (E).
Counting the losers as well as the winners costs one extra line and catches any cell that was misplaced.
7.SP.C.8Change Focus Count The ComplementTurn the GPA target into a point target first, then list every grade pair — the problem hides right on the boundary, where an A next to a C squeaks through but a B next to a C does not.
- Turn the GPA target into a point target
- Recover the missing C probabilities
- List all nine grade pairs
- Give each passing pair its probability
- Add the passing cells
- Cross-check with the failing cells