Probability and Counting: WRITE It Out

Here is a scene I watch play out every week. A sharp, busy professional is twenty minutes into the Quant section of the Executive Assessment. A counting question shows up. Their eyes light up, because they remember there is a FORMULA for this. Something with factorials. Something with a fraction stacked on a fraction. They write it down, plug in numbers, get an answer that matches one of the choices, and click.

It's wrong. And they never find out why, because there is nothing on their notepad to check.

Probability and counting questions do not punish people who are bad at math. They punish people who TRUST a formula they half remember. The fix is almost insultingly simple: write the cases out. A few of them. On paper. Every time.

Why this matters more on the EA than you think

The Quant section of the EA gives you 14 questions in 30 minutes. That's a little over 2 minutes per question, on average. The section mixes two formats. Problem Solving is a regular math question with five answer choices. Data Sufficiency gives you a question and two statements, and you decide whether the statements give you enough facts to answer it.

Each of the three sections is scored from 0 to 20, and the total runs from 100 to 200, with all three sections weighted the same. You can read the details on GMAC's page on the EA structure and scoring. With only 14 Quant questions, every single one carries real weight. Dropping a gettable counting question because you trusted a fuzzy formula is not a small leak. It's a hole in the boat.

So here's your forced choice. What do you want MORE: to feel clever using a formula, or to actually earn the point?

The two words you need first

Counting questions ask how many ways something can happen. Probability is the chance something happens, and on this test it almost always works out to this:

Probability = (number of outcomes you want) ÷ (total number of possible outcomes)

Both halves of that fraction are counts. So if you can count carefully, you can do probability. And the most careful way to count is not a formula. It's a list.

Try one before you read another word

That list is fifteen short numbers. Writing them takes well under a minute, and when you're done you KNOW the answer is right. The formula took less ink and gave you a wrong answer with total confidence. Which one was actually faster?

The 5 mistakes I see on these questions

1. Reaching for a formula you half remember. You learned combinations once, years ago. You sort of remember which number goes on the bottom. "Sort of" is not a strategy.
   INSTEAD: List the first few cases by hand. If a formula comes to mind, use the list to check it. If the two disagree, trust the list.
2. Counting in your head. You picture the outcomes, you tally them mentally, you get 4. It was 5. Nobody can check work that only exists inside your skull, including you.
   INSTEAD: Write every case as a short code. Pairs of dice as (2,6). People as letters. Then count the list twice.
3. Getting order wrong. Two dice show 3 and 5. Is that the same as 5 and 3? For dice, NO. Those are two different outcomes, because the dice are two different objects. For picking two people for a team, Ann and Ben is the same team as Ben and Ann.
   INSTEAD: Write both versions on paper and ask one question: "Is this a different result, or the same result written backwards?"
4. Ignoring "without replacement." That phrase means once you pull something out, it stays out. So the pool shrinks. People multiply the same fraction twice and walk straight into the trap.
   INSTEAD: Write the counts as they change. 6 red out of 10, then 5 red out of 9. Seeing the numbers drop on paper stops the mistake cold.
5. Listing EVERYTHING when the list is huge. Some people hear "write it out" and try to list 200 cases. That's not training, that's punishment.
   INSTEAD: List until the pattern shows up, like the 5, 4, 3 in the example above. Then finish with simple arithmetic. The list is there to show you the pattern and catch errors, not to replace thinking.

Quick sanity check with a classic. Roll two fair six-sided dice. What's the chance the sum is 8? Write it: (2,6), (3,5), (4,4), (5,3), (6,2). Five outcomes out of 36 total pairs, so 5/36. If you only wrote three of those because (3,5) and (5,3) "felt the same," you'd be wrong. Mistake number 3, caught by a list.

Now do it with Data Sufficiency

Data Sufficiency questions use the same five answer choices every time, so learn them cold:

• (A) Statement (1) ALONE is sufficient, but statement (2) alone is not.
• (B) Statement (2) ALONE is sufficient, but statement (1) alone is not.
• (C) BOTH statements TOGETHER are sufficient, but neither alone is.
• (D) EACH statement ALONE is sufficient.
• (E) The statements together are still NOT sufficient.

"Sufficient" means you can get exactly one answer. Here, listing cases turns into something we call TESTing VALUES: you plug in real numbers that fit the statement and see if the answer changes.

This is the same habit, everywhere

If this sounds familiar, it should. It's the same move as drawing every geometry figure, the same move as testing the edges on inequalities, and the same move as breaking exponents down to primes. Get the problem out of your head and onto paper, where you can see it and check it.

I'll bet that the last counting question you missed, you never wrote down a single case. Go look. If the page is blank, you just found your problem. And it's fixable.

Your challenge for today

Pull five probability or counting questions you've already done, including ones you got right. Redo every one of them with no formulas at all. List the first few cases, find the pattern, finish with arithmetic. Then put a mark next to any question where your list disagreed with your original answer. Every mark is a point you were giving away. Go find them.