🎲 Averages, Counting and Probability

Part 1 of 7 — Averages and Chance on the AR Subtest

Topic Overview

What These Questions Look Like

The Arithmetic Reasoning (AR) subtest is officially described as the "ability to solve basic arithmetic word problems." Averages show up as short stories about real numbers:

• Find the average: "A recruit's scores on four practice tests were 84, 90, 78 and 88. What was her average score?"
• Find the missing value: "What score does she need on the fifth test to raise her average to 86?"
• Combine two groups: "Twenty recruits average 80 on a test and thirty average 90. What is the average for all 50?"

Counting and probability questions tell a story too: how many different meal combos a menu allows, or the chance of pulling a certain name from a hat.

What is really being tested is whether you can turn the words into the right setup (add, divide, multiply) and carry it out cleanly by hand.

Basic probability and counting also appear on the Mathematics Knowledge (MK) subtest, whose official description lists probability. Everything in Parts 4 and 5 pays off twice.

The Conditions You Work Under

If you take a paper version of the test, the pace is much faster, so the shortcuts in this lesson matter even more.

AR is one of the four subtests in your AFQT score, the score that decides whether you can enlist, and it also feeds job line scores such as the Army GT score. Average questions follow a small set of patterns, which makes them some of the most winnable points on the subtest.

🎲 Averages, Counting and Probability

Part 2 of 7 — Finding the Average (Mean)

Add, Then Divide by How Many

The average (also called the mean) of a list of numbers is

$\text{average} = \frac{\text{sum of the values}}{\text{number of values}}$

Example 1. A part-time worker logged 7, 9, 8, 6 and 10 hours on five shifts. What was the average shift length?

Step 1 — Add in a column. $7 + 9 + 8 + 6 + 10 = 40$ hours.

Step 2 — Count the values. There are 5 shifts.

Step 3 — Divide. $40 \div 5 = 8$ hours.

Reasonableness: the shifts run from 6 to 10 hours, and 8 is right in that range. ✓

Key Fact: An average always lies between the smallest and largest values. If your answer is outside that range, you divided by the wrong count or made an addition slip.

🎲 Averages, Counting and Probability

Part 3 of 7 — Missing Values and Weighted Averages

The Score You Need on the Last Test

This is one of the most common average questions on AR. The trick is to think in totals, not averages:

$\text{needed score} = (\text{target average} \times \text{total count}) - (\text{sum so far})$

Example 1. A recruit's first four practice tests average 82. What score does she need on the fifth test to raise her average to 85?

Step 1 — Target total. To average 85 over 5 tests she needs $85 \times 5 = 425$ points.

Step 2 — Points so far. Four tests averaging 82: $82 \times 4 = 328$ points.

Step 3 — Subtract. $425 - 328 = 97$.

Balance check: each of the 4 tests is 3 points short of 85, so she is $4 \times 3 = 12$ points short in all. The fifth test must make up the 12: $85 + 12 = 97$. ✓

Reasonableness: 97 is possible on a 100-point test, and it is above 85, as it must be to pull the average up.

Example 2. Scores so far are 78, 84 and 90. What score on the fourth test gives an average of 85?

Target total: $85 \times 4 = 340$. So far: $78 + 84 + 90 = 252$. Needed: $340 - 252 = 88$.

Key Fact: The needed score is not "the target plus a little." Every earlier test that fell short adds to what the last test must make up.

🎲 Averages, Counting and Probability

Part 4 of 7 — Counting Outcomes

The Multiplication (Counting) Principle

If one choice can be made in $a$ ways and a second, separate choice in $b$ ways, the two choices together can be made in $a \times b$ ways. Add more choices by multiplying again.

Example 1 — outfits. A recruit has 5 shirts, 3 pairs of pants and 2 pairs of shoes. How many different outfits (one of each) are possible?

$5 \times 3 \times 2 = 30 \text{ outfits}$

Why multiply, not add? Each of the 5 shirts can go with each of the 3 pants: that is $5 \times 3 = 15$ shirt-and-pants pairs. Each of those 15 can go with either pair of shoes: $15 \times 2 = 30$. Adding ($5 + 3 + 2 = 10$) counts the items, not the outfits.

Example 2 — lock codes. A lock uses a 3-digit code, and each digit can be 0 through 9.

• Repeats allowed: each position has 10 choices: $10 \times 10 \times 10 = 1{,}000$ codes.
• No digit repeated: 10 choices for the first digit, then 9 left, then 8 left: $10 \times 9 \times 8 = 720$ codes.

Key Fact: Draw one blank for each position, write the number of choices in each blank, and multiply. If items cannot repeat, each blank gets one fewer choice than the one before.

🎲 Averages, Counting and Probability

Part 5 of 7 — Basic Probability

Favorable Over Total

When every outcome is equally likely,

$\text{probability} = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Write the fraction, then simplify it: answer choices are almost always in lowest terms.

Example 1. A standard die (faces 1 to 6) is rolled. What is the probability of rolling a number greater than 2?

Favorable: 3, 4, 5, 6, which is 4 outcomes. Total: 6 outcomes.

$\frac{4}{6} = \frac{2}{3}$

Example 2. A box holds 15 black pens, 9 blue pens and 6 red pens. One pen is picked without looking. What is the probability it is blue?

Total: $15 + 9 + 6 = 30$ pens. Favorable: 9 blue.

$\frac{9}{30} = \frac{3}{10}$

The denominator is the total of everything, not just the outcomes that are not favorable. Writing $\frac{9}{21}$ (blue over not-blue) is a common slip.

Key Fact: Every probability is between 0 and 1. A probability of 0 means the event cannot happen, and 1 means it is certain. An answer bigger than 1 is always wrong.

🎲 Averages, Counting and Probability

Part 6 of 7 — Test Strategy and Common Traps

The wrong answer choices on AR are not random. Each one is the result of a specific, common mistake. If you know the mistakes, you can see the traps coming.

The Seven Big Traps

Key Fact: Each of these traps produces a number that appears among the answer choices. Finding your answer in the list does not prove it is right. Your check does.

🎲 Averages, Counting and Probability

Part 7 of 7 — Mixed Mastery Set

This set mixes everything from Parts 2 through 5 the way the real AR subtest does: a few quick ones, several medium ones, and a few multi-step problems. Treat it like test day:

• No calculator. Use scratch paper for every sum and product.
• Read the last sentence first: average, total, missing value, count, or probability?
• Estimate before you compute (averages between low and high, probabilities between 0 and 1), and check before you answer, because on the real test you cannot go back.
• Aim for about 3.5 minutes per question or less.