Before you begin
This is the smallest family on the test and the one most often left out of a study plan. That combination is worth noticing. It is small, so the whole of it fits in an afternoon. It is left out, so most people arrive at those questions having done nothing about them. Both facts point the same way: this is the cheapest family on the test to get right.
It is also the family where the everyday word and the mathematical word sit closest together. When you say a thing is likely, you already mean something precise: that if the situation came round many times, it would happen in more of them than not. Probability just puts a number on that, and the number is a fraction you already know how to make.
Favorable over all
Here is the whole definition. Count the outcomes you are asking about. Count every outcome there is. Put the first over the second.
A bag holds twelve marbles. Three are red. You reach in without looking. The outcomes you are asking about — the favorable ones — number three. The outcomes there are number twelve. So the probability is 3/12, which simplifies to 1/4.
The word favorable is doing no moral work. It does not mean good. It means the outcomes that count as the thing you asked about. If the question is about drawing a red marble, red is favorable. If the question is about not drawing a red marble, the other nine are.
Three ways to write the same number
A probability of 1/4 is also 0.25 and also 25%. They are the same number in three costumes, and this is exactly the work of family 5. The test will take any of the three, and its multiple-choice answers will often be given in a form you did not expect, so being able to move between them quickly is most of what makes these questions fast.
To go from the fraction: divide. 1 ÷ 4 = 0.25. To go from the decimal to the percent: the decimal point moves two places right. 0.25 becomes 25%.
Zero, one, and the room between
A probability cannot be less than zero and cannot be more than one. Nothing can happen fewer than no times out of the total, and nothing can happen more times than the total.
That gives you a check you can run on every answer in this family. If you have calculated a probability of 1.4, or of 140%, you have not made a small arithmetic slip: you have put the wrong number on the bottom. Go back and count the outcomes again.
Zero means impossible. Drawing a green marble from a bag with no green marbles in it is a probability of 0. One means certain. Drawing a marble from a bag of twelve marbles is a probability of 1, or 12/12. Everything real sits between.
The other side: what does not happen
Something either happens or it does not. There is no third option. So the probability that it happens and the probability that it does not must add up to the whole thing — to 1, or to 100%.
That gives you the second tool in this family, and it saves more time than any other. If the forecast says a 30% chance of rain, the probability of no rain is 100 − 30 = 70%. You did not need to know anything about clouds. You subtracted.
The same works with fractions. If the probability of red is 3/12, the probability of not red is 1 − 3/12 = 9/12, which is 3/4. You can also just count the nine that are not red, and you should, because getting the same answer two ways is the best check there is.
Two things happening together
Flip a coin twice. What is the probability of heads both times?
You can list every outcome. There are four: HH, HT, TH, TT. One of them is two heads. So the probability is 1/4.
Listing works and you should use it when the numbers are small, because it is impossible to get wrong. But there is a shorter way, and it is the one the test expects: when two things both have to happen, multiply their probabilities. The chance of heads is 1/2. The chance of heads again is 1/2. And 1/2 × 1/2 = 1/4.
Multiplying makes the answer smaller, which is right. Asking for two things to go your way is asking for more than asking for one.
One caution. The multiplying rule as written assumes the second event is unaffected by the first — the coin has no memory. If you draw a marble from the bag and keep it, the bag has changed and the second fraction has eleven on the bottom, not twelve. The GED asks the simpler kind far more often, but read the question for the word replace or the phrase and put it back, because it tells you which one you are in.
Counting the ways
The last part of this family is not about likelihood at all. It is about how many different ways a set of choices can be made, and it turns out to be multiplication again.
You own four shirts and three pairs of trousers. How many outfits? For each of the four shirts there are three choices of trousers, so 4 × 3 = 12. You could draw it as a tree with four branches, each splitting into three, and count the ends: twelve.
This is called the counting principle, and it is multiplication with a name. When one choice follows another, and each choice is free of the others, multiply the number of options. A diner with 4 sandwiches, 3 soups and 2 drinks offers 4 × 3 × 2 = 24 lunches.
When the order matters, and when it does not
The GED’s own list of what it tests names two more words in this family: permutations and combinations. They sound technical. They are the counting principle again, with one question added to it.
Start with something small enough to list. Four people — Ana, Ben, Cal and Dee — and two of them are needed.
Suppose the two jobs are different: one person runs the meeting and the other takes the notes. Then Ana running it with Ben on notes is not the same as Ben running it with Ana on notes; the same two people, doing different things. Any of the 4 could run the meeting, and once someone is, any of the 3 left could take notes. That is 4 × 3 = 12. When the order of the choosing makes a difference like this, each way of choosing is called a permutation.
Now suppose the two are just a pair — two people to carry a table, and nobody cares who takes which end. Then Ana-and-Ben is the same pair as Ben-and-Ana. List them and there are only six: AB, AC, AD, BC, BD, CD. When the order makes no difference, each way of choosing is called a combination.
| The pair | The same pair, counted with order |
|---|---|
| Ana and Ben | AB · BA |
| Ana and Cal | AC · CA |
| Ana and Dee | AD · DA |
| Ben and Cal | BC · CB |
| Ben and Dee | BD · DB |
| Cal and Dee | CD · DC |
| 6 combinations | 12 permutations |
Twelve and six, and the table shows why: every pair appears twice among the twelve, once each way around. So you get the combinations by counting the permutations and then dividing by the number of ways the chosen ones could be put in order — here, 2.
The same thing, bigger
Eight runners in a race, and medals for first, second and third. Order matters — gold is not bronze. Any of the 8 could win gold; then any of the 7 left could take silver; then any of the 6 left, bronze. That is 8 × 7 × 6 = 336 ways the medals could go.
Now eight people, and a committee of three to be chosen from them. Order does not matter — nobody on a committee is first. Every group of three was counted in that 336 once for each way of lining those three people up, and three people can be lined up in 3 × 2 × 1 = 6 ways. So the number of committees is 336 ÷ 6 = 56.
That 3 × 2 × 1 has a name, the factorial, and a sign, the exclamation point: 3! means 3 × 2 × 1 = 6, and 4! means 4 × 3 × 2 × 1 = 24. It is simply the number of ways to put that many things in a row.
The calculator does the arithmetic
The TI-30XS they give you has both of these on its PRB key: nPr for permutations and nCr for combinations. For the medals: 8, PRB, nPr, 3, enter, and it says 336. For the committee: the same with nCr, and it says 56. There is a factorial key in the same menu. The calculator page has the keystrokes.
The formula sheet does not give a formula for either one. That is a fair hint about what the test cares about: not the arithmetic, which the machine will do, but the question in the box above, which only you can answer.
For the record, since prep books print them: the formulas are nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!), where n is how many there are to choose from and r is how many you choose. You do not need them. Multiplying down and dividing, the way this section does it, gets the same answers, and you can see why each step is there.
The guard
Three checks before you write an answer down.
First, is your probability between 0 and 1? If it is not, you counted the total wrong.
Second, did the question ask for a probability or for a count? “How likely” wants a fraction between 0 and 1. “How many ways” wants a whole number, usually a biggish one. They use the same multiplication and they want different kinds of answer, and that is the one confusion this family reliably produces.
Third, when the question asks how many ways, swap two of the chosen things in your head. If that makes a different outcome, multiply down. If it does not, multiply down and then divide by the number of ways to put the chosen ones in order.
Four to study before you start
A drawer holds 5 black socks, 3 white and 2 gray. You take one without looking. What is the probability it is white?
Count the favorable outcomes: 3 white. Count every outcome: 5 + 3 + 2 = 10 socks. So the probability is 3/10. As a decimal that is 3 ÷ 10 = 0.3, and as a percent, 30%. All three are correct, and the test may offer any of them.
From the same drawer, what is the probability the sock is not gray?
Two ways, and they must agree. Count what is not gray: 5 black + 3 white = 8, out of 10, so 8/10 = 4/5. Or take the gray probability from the whole: gray is 2/10, and 1 − 2/10 = 8/10. Same answer. When a question says not, the subtraction is usually quicker, and the counting is the check.
A fair six-sided die is rolled twice. What is the probability of a 6 both times? And how many different two-roll results are there altogether?
Both times is a multiplication: 1/6 × 1/6 = 1/36. For the count, the first roll has 6 outcomes and so does the second, so there are 6 × 6 = 36 results in all — which is where the 36 on the bottom of that fraction came from. The two questions are the same arithmetic asked from two directions.
A tenants’ association has 7 members on its board. In how many ways can it choose a chair, a secretary and a treasurer? And in how many ways can it choose three of the seven to go to a meeting with the landlord?
Ask the question first. For the officers, swap two people and you have a different outcome — the chair and the treasurer are different jobs — so order matters. Multiply down: 7 × 6 × 5 = 210.
For the meeting, swap two people and nothing changes; the same three people are going. Order does not matter, so divide the 210 by the ways of putting three people in order, 3! = 6: 210 ÷ 6 = 35.
Same seven people, same arithmetic until the last step, and the thing that decides the last step is in the wording: named jobs, or just a group. On the calculator, 7 nPr 3 gives 210 and 7 nCr 3 gives 35.
Now you
Work on paper, without a calculator. Give probabilities as fractions in lowest terms unless the question asks otherwise. Questions 6 and 10 have two parts.
- A bag holds 4 red marbles, 6 blue and 2 green. You draw one without looking. What is the probability it is blue?
- From the same bag, what is the probability the marble is green?
- A weather forecast gives a 45% chance of snow. What is the probability it does not snow?
- You roll one fair six-sided die. What is the probability of rolling a number greater than 4?
- A) 1/6
- B) 1/3
- C) 1/2
- D) 2/3
- A spinner is divided into 8 equal parts, 3 of them shaded. Write the probability of landing on a shaded part as a percent, to the nearest whole percent.
- A cafe offers 5 sandwiches, 4 soups and 3 drinks. (a) How many different sandwich-soup-drink lunches can be ordered? (b) If one lunch is chosen at random from all of them, what is the probability it is the one you wanted?
Reading A reading rest stop, for the stubborn ones.
Ten minutes with the last section of this page — The Company, below the answer key — before you finish the quiz. It is about the first person who counted deaths in London on purpose, and found that chance has a shape. It will not help you with question 7. It may change what you think these fractions are for.
No picture at this rest stop either. The bag of marbles at the start of the Guide is the only drawing in this quiz, and everything below uses it.
- A coin is flipped twice. What is the probability of getting tails both times?
- Marisol says that if a bag holds 3 red and 7 blue marbles, the probability of red is 3/7. Is she right? If not, say what the probability is and what she has confused it with.
- A box holds 20 bulbs and 2 of them are dead. You take one bulb at random. What is the probability it works? Give your answer as a percent.
- A) 2%
- B) 10%
- C) 18%
- D) 90%
- A bag holds 5 white counters and 3 black. (a) You draw one, look at it, and put it back. Then you draw again. What is the probability both are white? (b) Instead you draw one and keep it, then draw a second. What is the probability both are white now?
Check your work
The man who counted the dead
In 1662 a London haberdasher named John Graunt published a small book with a long title: Natural and Political Observations Made upon the Bills of Mortality. He was not a mathematician. He sold buttons and cloth. What he had was patience and a pile of paper nobody else had thought worth reading.
The Bills of Mortality were weekly lists. Every parish in London recorded its burials and what had killed each person, and the lists were printed and sold so that the wealthy could tell when plague was rising and it was time to leave for the country. People read them the way you read a weather forecast: for this week only, then thrown away.
Graunt kept them. He gathered decades of them and did something that had not been done: he added them up and looked for what stayed the same.
What he found should have been impossible. The individual deaths were the most unpredictable events imaginable — who would die this week, and of what, was knowable to nobody. Yet the totals held steady. Roughly the same number died of the same causes year after year. The ratio of boys born to girls was close to constant, and slightly favored boys, every single year. Chance, looked at one case at a time, was chaos. Looked at ten thousand cases at a time, it had a shape.
That is the whole idea underneath the fractions in this quiz. A single draw from the bag tells you nothing; you cannot predict it and neither can anyone else. But 3 out of 12 is a genuine claim about what happens when the draw is repeated, and it is as reliable as any measurement.
Graunt went further. From the burial lists he built the first life table — an estimate of how many of a hundred people born would still be alive at six, at sixteen, at fifty-six. It was rough and he knew it. It was also the first time anyone had written down the odds of an ordinary life, and it is the direct ancestor of every insurance policy and every public health statistic since.
He was made a Fellow of the Royal Society on the strength of it, at the personal insistence of Charles II, who reportedly said that if they found any more shopkeepers like him they should admit them all without further discussion.
Two things are worth carrying from this. The first is that the mathematics in this family was not invented in a university. It was invented by someone reading a public record carefully, because the record was about people and he wanted to know what it said.
The second is what happened to the Bills of Mortality afterward. Once it was understood that the numbers held a shape, the shape became an argument. Nineteenth-century reformers used exactly Graunt's method to show that the poor of one district died younger than the comfortable of another, by amounts far too large and far too steady to be chance. That is still what these fractions are for. A probability is a claim about what happens when a thing is repeated, and some things are repeated to some people far more often than to others.