Counting and Probability Problems

Probability on Arithmetic Reasoning is deliberately modest: one or two events, whole numbers, and answers that come out as clean fractions. The difficulty is almost entirely in reading whether the events are sequential or alternative, and whether the pool changes.

Simple probability

Probability = favorable outcomes / total outcomes.

A box holds 5 red, 8 blue and 7 green markers. One is drawn at random. What is the probability it is blue?

Total: 5 plus 8 plus 7 is 20. Favorable: 8.

Probability: 8/20, which reduces to 2/5, or 0.4, or 40 percent.

Three things to make automatic:

Count the whole pool, not just the category mentioned. Writing 8/15 - blue over the non-blue - is the most common error here and it is always a choice.

Reduce the fraction. Answer choices are in lowest terms.

Check the range. Every probability is between 0 and 1. An answer above 1 is a setup error, not a hard question.

The complement

The probability that something does not happen is 1 minus the probability that it does. Not blue is 1 minus 2/5, which is 3/5.

Reach for this whenever a question says "at least one". The chance of at least one is much easier as 1 minus the chance of none.

And, or, and the pool

Enlarge

Zoom in

From the same box of 20 markers, two are drawn. What is the probability both are red?

Everything depends on one question the problem must answer: is the first marker put back?

With replacement, the pool is unchanged: 5/20 times 5/20, which is 1/16.

Without replacement, the second draw sees 4 reds among 19 markers: 5/20 times 4/19, which is 1/4 times 4/19, or 1/19.

Word problems usually mean without replacement, because people do not usually put things back, but the question will say. Look for "replaced", "returned to the box", or "without replacement". When it does not say and the items are physical objects being taken, assume they are not replaced.

Or, and the overlap

One card is drawn from a deck of 52. What is the probability it is a heart or a face card?

Hearts: 13. Face cards: 12. But three cards are both - the jack, queen and king of hearts - and adding 13 plus 12 counts them twice.

13 plus 12 minus 3 is 22, so 22/52, which reduces to 11/26.

Subtract the overlap whenever the two categories can both be true. If they cannot - drawing a red card or a spade - there is no overlap and you just add.

Counting arrangements

A uniform can be assembled from 4 shirts, 3 pairs of trousers and 2 pairs of boots. How many combinations?

Multiply the choices at each stage: 4 times 3 times 2 is 24.

That rule covers this subtest. Multiply the number of options at each independent stage.

When items are chosen from the same pool and not repeated, the pool shrinks at each stage:

Four people are available and three will be assigned to distinct positions - team lead, driver and radio operator. How many ways?

Four choices for the first position, three for the second, two for the third: 4 times 3 times 2 is 24.

If the three roles were interchangeable - just "pick three people" - the order would not matter and the count would be smaller. Arithmetic Reasoning generally asks the ordered version, with distinct roles, because the unordered version needs a combination formula that belongs to Mathematics Knowledge.

Probability expressed as odds

Occasionally a question gives odds rather than a probability, and they are not the same number.

Odds of 3 to 1 in favor means 3 favorable to 1 unfavorable, which is 4 total, so the probability is 3/4 - not 3/1 and not 1/3.

This is the ratio-is-not-a-fraction problem from ratio and proportion, wearing different clothes. Add the parts.

What you can skip

Across the 66 questions on this topic:

• Odds. The section above explains them because many study guides do, but "odds" never appears in an Arithmetic Reasoning question in our bank (or a Mathematics Knowledge one). Skim it.
• Combination formulas. "Combination" appears six times, always meaning "how many outfits or codes", which the multiplication rule answers.

Where people lose points

A denominator that is not the whole pool. The usual error.

Not adjusting the pool for a second draw, when the first item was kept.

Adding when the events are sequential. Both happening is a product. The sum is usually larger than either probability, which is impossible for an "and", so checking the size catches it.

Double-counting the overlap on an "or" question.

Leaving the fraction unreduced and concluding your answer is not a choice.

Reading odds as a probability.

Work one in under a minute

A drawer holds 6 pairs of black socks and 4 pairs of gray. Two pairs are taken at random without looking. What is the probability both are gray?

First draw: 4 gray out of 10, which is 2/5.

Second draw: 3 gray out of the 9 left, which is 1/3.

Both: 2/5 times 1/3 is 2/15.

Check: 2/15 is about 0.13, which is well under the 0.4 chance of the first one being gray, as it must be.

Where this leads

The counting rule and the and-or distinction are the same on Mathematics Knowledge, which asks them without a story and adds combinations.