Chance in words and the 0 to 1 scale
One person says rain is “likely”, another says “probably not”, a third gives a percent. How can chance be put on one scale, so that everyone means the same thing and two chances can be compared?
Where you are 14 in Basic probability
Why this exists
People talk about chance mostly in words: “no way”, “probably not”, “fifty-fifty”, “probably”, “definitely”. Words are loose. When one person says rain is “likely”, do they mean more likely than not, or nearly sure? Two people can use the same word and mean different things.
What randomness means showed that a random event cannot be called in advance, yet follows a steady pattern over many tries. This lesson gives that pattern a ruler: a line from 0 to 1, with the everyday words given exact places on it. Once a chance sits on the line, two chances can be compared without arguing about words.
Definition
An event is something that might or might not happen: it rains tomorrow, the school bus is late, a raffle ticket drawn without looking is yours. Its exact meaning, built from a list of outcomes, is in outcomes and events.
The probability of an event is a number from 0 to 1 that says how likely the event is. A larger number means a more likely event.
| Word | Probability | What it says |
|---|---|---|
| impossible | 0 | cannot happen |
| unlikely | between 0 and \tfrac{1}{2} | less likely to happen than not |
| even chance | \tfrac{1}{2} | as likely to happen as not |
| likely | between \tfrac{1}{2} and 1 | more likely to happen than not |
| certain | 1 | must happen |
Even chance is also called “fifty-fifty”. “Very unlikely” and “very likely” are used near 0 and near 1; they have no exact border.
A probability may be written as a fraction, a decimal or a percent, and all three name the same point: \tfrac{1}{2} = 0.5 = 50\%. As percents the scale runs from 0\% to 100\%. Changing forms is the work of writing a fraction as a decimal and percent as a rate per hundred.
Intuition
Figure 1 is a ruler for chance. The further right an event sits, the more likely it is.
The middle point, \tfrac{1}{2}, is the tipping point: to its left an event is less likely to happen than not, to its right more likely. The two ends are the only places where you know in advance what will happen: at 0 it will not, at 1 it will.
So a probability can be pictured as the event’s share of the whole chance: none of it at 0, all of it at 1, half at \tfrac{1}{2}. A weather forecast gives another picture: if the forecaster is well calibrated, then on the days given a 70\% chance of rain it rains on about 7 in every 10, and reading a probability as a share of many repeats like this is experimental probability.
Results
The scale. If p is the probability of an event, then
As a percent, no probability is below 0\% or above 100\%.
The ends. A probability is 0 exactly when the event is impossible, and 1 exactly when it is certain. An event that can happen but need not has 0 < p < 1.
Larger means more likely. One event is more likely than another exactly when its probability is larger. To compare probabilities given in different forms, write both in one form first.
An event that comes with another. If event B happens every time event A happens, then the probability of A is at most the probability of B.
Why it is true
Count the whole chance as 1, and read a probability as the share of it that belongs to the event.
The scale. A share cannot be less than none of the whole or more than all of it, so it lies from 0 to 1.
The ends. An impossible event has none of the chance, so its probability is 0; a certain event has all of it, so its probability is 1. In every situation in this course each possible result has some share of the chance, so an event that can happen has a share above 0, and an event that can fail leaves some of the chance to failing, so its share is below 1.
Even chance is the middle. “As likely to happen as not” means happening and not happening have equal shares. Together they make up the whole, 1, so each is \tfrac{1}{2}.
Larger means more likely. A larger probability is a larger share of the chance, which is what “more likely” means.
An event that comes with another. A never happens without B, so all of the chance that A happens is also chance that B happens. B has at least A’s share, and may have more of its own.
Worked examples
Choosing the word
Describe each event with one of the words impossible, unlikely, even chance, likely or certain.
- (a) In a town far from the North and South Poles, the sun rises tomorrow.
- (b) A raffle has 200 tickets, well mixed in a drum, and Year 7 bought exactly 100 of them. The winning ticket is a Year 7 ticket.
- (c) In the same raffle, the winning ticket is ticket number 1.
(a) Certain, probability 1. Far from the poles the sun rises every day of the year.
(b) Even chance, probability \tfrac{1}{2}. The tickets are well mixed, so each is as likely to win as any other, and Year 7 holds exactly as many as everyone else. A Year 7 winner is as likely as not.
(c) Unlikely, in fact very unlikely. Ticket 1 is one ticket, and 199 others are each just as likely to win. It can happen, so the probability is above 0, but it is close to 0.
From numbers to words
Five events have these probabilities: A, 0.1; B, 0.35; C, 50\%; D, \displaystyle \frac{3}{4}; E, 95\%. Place them on the probability scale and describe each in words.
SolutionWrite all five in one form so they can be compared with 0, \tfrac{1}{2} and 1. As decimals, 50\% = 0.5, \displaystyle \frac{3}{4} = 0.75 and 95\% = 0.95.
- A: 0.1 is less than 0.5 and close to 0, so A is unlikely, indeed very unlikely.
- B: 0.35 is less than 0.5, so B is unlikely, but nearer the middle than A.
- C: 0.5 is exactly \tfrac{1}{2}, so C has an even chance.
- D: 0.75 is more than 0.5, so D is likely.
- E: 0.95 is more than 0.5 and close to 1, so E is very likely, but not certain, since 0.95 is not 1.
Reading Figure 2 from left to right lists the events from least to most likely: A, B, C, D, E.
Two forecasts in different forms
Jessamy is planning a picnic for one of three days. The sites she checks write the chance of rain in different forms: Saturday 40\%, Sunday \displaystyle \frac{3}{5}, Monday 0.35. Order the days from least to most likely to be rainy, say which day suits the picnic best, and describe each day’s chance of rain in words.
SolutionPut the three numbers in one form. As decimals, 40\% = 0.40 and \displaystyle \frac{3}{5} = \frac{6}{10} = 0.6; Monday is already 0.35.
Now compare the decimals: 0.35 < 0.40 < 0.6. From least to most likely to be rainy, the days are Monday, Saturday, Sunday. A smaller probability means a less likely event, so Monday is the best bet for a dry picnic.
In words, Monday and Saturday are both left of \tfrac{1}{2}, so rain is unlikely on each; on Sunday rain is likely. Monday and Saturday get the same word but different numbers: the word alone could not choose between them, and the numbers can.
A bag known only by its largest color
Kasper has a bag of green, yellow and white marbles, with at least one of each. All he knows about the numbers is that more than half of the marbles are green. He draws one without looking. Describe the chance that it is (a) green, (b) yellow, (c) purple. (d) Can he say that yellow is very unlikely?
Solution(a) Likely. Each marble is as likely to be drawn as any other, and more than half of them are green, so green has more than half the chance: its probability is above \tfrac{1}{2}.
(b) Unlikely. Green alone is more than half the bag, so yellow and white together are fewer than half, and yellow alone fewer still. Its probability is below \tfrac{1}{2}, and above 0 since there is at least one yellow marble.
(c) Impossible, probability 0. There are no purple marbles.
(d) No. What he knows fixes which side of \tfrac{1}{2} each color is on, not how far from the middle. A bag of 51 green, 48 yellow and 1 white fits the description, and so does 51 green, 1 yellow and 48 white. In the first, yellow is almost as likely as not; in the second, it hardly ever comes out. “Unlikely” is as far as the information goes.
A ladder of birthday events
A person is chosen at random from a large crowd. Take births to be spread about evenly over the days of the year. Order these events from least to most likely, and describe each in words. The person’s birthday falls on:
- A: the 32nd of a month;
- B: the 31st of a month;
- C: the 30th of a month;
- D: a day numbered 1 to 28;
- E: a day numbered 1 to 31.
The ends first. A is impossible, since no month has a 32nd: its probability is 0. E is certain, since every month has from 28 to 31 days, so every date is numbered 1 to 31: its probability is 1.
D leaves out only the 29ths, 30ths and 31sts, at most three days a month, so it is likely, but not certain.
B and C each pick out at most one day a month, so both are unlikely. To compare them, look at the months. The seven months with a 31st (January, March, May, July, August, October, December) all have a 30th too, and April, June, September and November have a 30th but no 31st. So every 31st is matched by a 30th in the same month, with four 30ths left over, and with births spread evenly C is more likely than B.
The order is A, B, C, D, E: impossible, unlikely, unlikely (a little less so), likely, certain. As a check, E happens every time D does, so by the last result D cannot be more likely than E, as found.
Never seen is not impossible
Iestyn keeps a record of his school bus. On the last 20 dry, clear mornings it was never late. On the last 20 snowy mornings it was late every time. Is a late bus on a clear morning impossible? Is a late bus on a snowy morning certain? Place each roughly on the scale.
SolutionNeither. A bus can be late on a clear day: it can break down, or a road can be closed. Twenty mornings without a late bus show that lateness is rare then, not that it cannot happen; and on some snowy morning the roads may be cleared early.
So a late bus on a clear morning is very unlikely, close to 0 but not 0, and a late bus on a snowy morning is very likely, close to 1 but not 1. Turning a record like this into numbers is experimental probability.
Boundaries
Only three words name a single number. Impossible is exactly 0, even chance exactly \tfrac{1}{2}, certain exactly 1. Unlikely and likely each cover a stretch. A probability of 0.48 is, strictly, on the unlikely side, though everyday speech would call it “about even”. When a word is not precise enough, use the number.
Impossible and certain depend on the situation. The sun rising tomorrow is certain far from the poles, but well inside the Arctic Circle there is a stretch of weeks in midwinter when it does not rise at all. A probability belongs to an event in a stated situation.
A probability does not say what happens next. An event with probability 0.9 can fail and one with probability 0.1 can happen. A single result does not show a probability was wrong.
Two possible results do not make an even chance. The bus is late or not, but that does not make lateness \tfrac{1}{2} likely. Even chance needs the two to be equally likely, which is the question of fair and unfair chance devices.
Probability 0 and impossible match only when there is a list of possible results, as in every situation in this course. With infinitely many possible results, such as a point chosen anywhere on a line segment, each exact point has probability 0, yet some exact point is always chosen.
Where this goes next
Here events are placed on the scale by reasoning. Next comes the exact number: once the outcomes are listed and an event picked out, the probability of an event compares how many outcomes make it happen with how many there are.
An event and its not happening have probabilities that add to 1, which is complements. Whether outcomes are equally likely is probability models; records like Iestyn’s lead to experimental probability and the law of large numbers; and odds compare happening with not happening.
Exercises
Choosing the word
Describe each event with one of the words impossible, unlikely, even chance, likely or certain, and say why.
- A jar holds only green buttons. Gideon takes one without looking. Describe the chance that it is green, and the chance that it is red, and give the probability of each.
- A person chosen at random from a large crowd has a birthday on the 1st of a month.
- A person chosen at random from a large crowd has a birthday on a day numbered 1 to 30.
- On snowy mornings, Marisol’s bus is late exactly as often as it is on time, in the long run. It is late on the next snowy morning.
- In a town far from the poles, the sun rises in the west tomorrow.
From numbers to words
In Exercises 6 to 9, describe an event with the given probability using one of the words impossible, unlikely, even chance, likely or certain.
- 0.15
- \displaystyle \frac{5}{8}
- 55\%
- \displaystyle \frac{6}{12}
- Which of these numbers could not be the probability of any event: 1.3, 0, -0.4, \displaystyle \frac{7}{8}, 120\%, \displaystyle \frac{9}{8}?
Comparing chances
- The forecast gives a 0.35 chance of rain on Thursday and a 30\% chance on Friday. On which day is rain more likely?
- Order these probabilities from the least likely event to the most likely: \displaystyle \frac{2}{5}, 0.55, 48\%, \displaystyle \frac{1}{4}.
- In bag X, more than half the marbles are red. In bag Y, exactly half are red. A marble is drawn from each without looking. From which bag is a red marble more likely?
- A person is chosen at random from a large crowd, with births spread about evenly over the year. Which is more likely, a birthday on the 29th of a month or on the 31st? Explain without working out either probability.
- For Leontine’s bus, being late on a snowy morning is likely, and being late on a clear morning is unlikely. Which of the two probabilities is larger, and which number on the scale must lie between them?
Reasoning
- Leontine says, “The sun rising tomorrow is certain, everywhere on Earth.” Explain why that is not right for a town well inside the Arctic Circle in midwinter.
- The forecast said 80\% chance of rain, and the day stayed dry. Gideon says the forecast was wrong. Explain why one dry day does not show that.
- Kasper says that a probability of 0.000001 is the same as impossible. Explain why it is not.
- Explain, reading a probability as the event’s share of the whole chance, why no event can have a probability greater than 1 or less than 0.
- Jessamy says, “Either the bus is late or it is not. That is two possibilities, so the chance it is late is \tfrac{1}{2}.” Explain what is wrong with her reasoning.
Answers
- Green is certain, with probability 1: every button in the jar is green. Red is impossible, with probability 0: there are no red buttons to take.
- Unlikely. Each month has only one 1st, so most days of the year are not a 1st. It can happen, so the probability is above 0, but it is close to 0.
- Likely, in fact very likely. Only 31sts are left out, at most one a month, so nearly every day of the year is numbered 1 to 30. It is not certain, because a 31st is possible.
- Even chance, probability \tfrac{1}{2}. Late exactly as often as on time means as likely to happen as not.
- Impossible, probability 0. The sun always rises in the east.
- Unlikely. 0.15 < 0.5, and it is fairly close to 0.
- Likely. \displaystyle \frac{5}{8} = 0.625, which is more than 0.5.
- Likely. 55\% = 0.55, which is more than 0.5, though only just.
- Even chance. \displaystyle \frac{6}{12} = \frac{1}{2}.
- 1.3, -0.4, 120\% and \displaystyle \frac{9}{8}. A probability lies from 0 to 1: 1.3 and \displaystyle \frac{9}{8} = 1.125 are above 1, -0.4 is below 0, and 120\% = 1.2 is above 1. Both 0 and \displaystyle \frac{7}{8} are on the scale.
- Thursday. 30\% = 0.30, and 0.35 > 0.30.
- \displaystyle \frac{1}{4}, \displaystyle \frac{2}{5}, 48\%, 0.55. As decimals they are 0.25, 0.4, 0.48 and 0.55.
- Bag X. There, red has a probability above \tfrac{1}{2}; in bag Y it is exactly \tfrac{1}{2}.
- The 29th. Every month that has a 31st also has a 29th, and so do April, June, September and November, and February in a leap year. So a year has more 29ths than 31sts, and with births spread evenly a 29th is more likely.
- Late on a snowy morning: its probability is above \tfrac{1}{2}, and late on a clear morning is below \tfrac{1}{2}. So \tfrac{1}{2} lies between them.
- Well inside the Arctic Circle there are weeks in midwinter when the sun does not rise at all. In the middle of those weeks the sun rising tomorrow is not certain; it is impossible. Whether an event is certain depends on the situation it is described in.
- A chance of 0.8 is likely, not certain, so the day can still be dry. If the forecaster is well calibrated, then over many days given an 80\% forecast it rains on about 8 in 10 and stays dry on about 2, and this day can be one of those 2. Only a long record of such forecasts could show that they were off.
- Impossible means the event cannot happen, and its probability is exactly 0. A probability of 0.000001 is above 0: the event is very unlikely, but it has some share of the chance, so it can happen.
- The whole chance counts as 1. An event’s share of it can be none of it (0), all of it (1) or a part in between, but never less than none or more than all. So every probability lies from 0 to 1: a number such as 1.2 would be more than the whole chance, and -0.3 less than none of it.
- Having two possible results does not make them equally likely. Even chance means late as often as on time, and that depends on the bus and the weather: on a clear morning lateness may be very unlikely, on a snowy one very likely. Counting the possibilities says nothing about how likely each one is.