Probability of an event

When you reach for it

The words of the chance scale say roughly how likely something is. Often you can give the exact number instead. A prize is taped under one of fifty seats, and your family sits in three of them. A bag holds twelve counters, five of them red. A shuffled playlist picks its first song from forty. In each case you know every result that can happen, you know which ones you are hoping for, and nothing makes one result come up more often than another.

That last condition is the one this lesson needs. When every outcome in the sample space is equally likely, the probability of an event is found by counting: how many outcomes make the event happen, out of how many there are. The words outcome, sample space and event are the ones from sample spaces, outcomes and events: an outcome is one single result, the sample space S is the set of all of them, and an event is any collection of outcomes.

Over two days: Part 1 counts and writes the answer three ways; Part 2 counts in two stages, compares events, and works backwards from a probability to a count.

The procedure

Part 1: counting the outcomes in an event

The rule. If every outcome in the sample space S is equally likely, then for any event A

P(A) is read “the probability of A”. The top counts the outcomes that make A happen, and the bottom counts every outcome there is.

Why the rule is right. On every trial something in S happens, so the whole sample space has probability 1, the top of the chance scale. Suppose S has n equally likely outcomes. Nothing gives any one of them a bigger share, so they split that 1 equally and each has probability \displaystyle \frac{1}{n}. A trial gives exactly one outcome, so two outcomes never happen together and their shares never overlap. An event A with k outcomes in it therefore has k shares of \displaystyle \frac{1}{n}, which is \displaystyle \frac{k}{n}.

One outcome or several. A single outcome is the case k = 1: with 50 seats, the prize is under one particular seat with probability \displaystyle \frac{1}{50}. An event with several outcomes just has a larger top. If your family sits in 3 of the 50 seats, the event “the prize is under one of our seats” has probability \displaystyle \frac{3}{50}.

Count the equally likely things. Nine letter tiles spelling TENNESSEE are equally likely to be drawn, but the letters are not: E is on 4 tiles and T on 1. So the count is over the 9 tiles, never over the 4 letters (Example 2).

The answer is always from 0 to 1. The top is as small as 0 when no outcome fits the event, and as large as the bottom when every outcome fits. So

with P(A) = 0 for an impossible event and P(A) = 1 for a certain one, as on the chance scale.

Writing a probability three ways

A probability is one number, and it can be written as a fraction, a decimal or a percent. Take the event “the prize seat is a multiple of 6”: 8 of the 50 seats fit, so the probability is \displaystyle \frac{8}{50}.

As a fraction. Dividing top and bottom by the same number gives an equivalent fraction: \displaystyle \frac{8}{50} = \frac{4}{25}. Both are correct; the unsimplified one still shows the two counts.

As a decimal. A fraction is a division (fraction to decimal): 8 \div 50 = 0.16. Some divisions never stop, such as \displaystyle \frac{4}{9} = 0.444\ldots, written 0.\overline{4} with a bar over the digit that repeats. For use, round it: \displaystyle \frac{4}{9} \approx 0.444.

As a percent. Percent means “out of 100”, so multiply the decimal by 100 (fraction to percent): 0.16 is 16 hundredths, which is 16\%. When the bottom divides 100, an equivalent fraction over 100 shows the same thing: \displaystyle \frac{8}{50} = \frac{16}{100} = 16\%.

So

The fraction is exact and shows where it came from, the decimal is the easiest form to compare, and the percent is the everyday one. The range 0 to 1 becomes 0\% to 100\%.

Part 2: two stages, comparing events, and working backwards

Experiments in two stages. Drawing a counter and then another, or a shuffle choosing the first two songs, has outcomes that are pairs. List them in a table with a row for each result of the first stage and a column for each result of the second, as in listing a sample space; each cell is one outcome. The cells are equally likely when the first results are equally likely, every row has the same number of cells, and, whatever happened first, the second results that can still happen are equally likely. Then count cells exactly as for a single draw.

Comparing two events. The event with the larger probability is the more likely one. Two events from the same sample space have fractions with the same bottom, so comparing them means comparing their tops. Events from different sample spaces, such as two bags, need the probabilities themselves: write both as decimals, or put both over a common denominator. A bag with more red counters need not give the better chance of red; what matters is the share of the bag that is red.

Working backwards. The rule ties three numbers together: the probability, the number of outcomes in the event and the number in the sample space. Knowing two gives the third. Multiplying both sides of \displaystyle P(A) = \frac{k}{n} by n gives

and, when P(A) is not 0, dividing by P(A) gives n = k \div P(A). This is solving a proportion: the event’s count is to the whole as the probability is to 1. Both k and n count objects, so both must be whole numbers; if the arithmetic does not come out whole, no bag of that size has that probability.

Looking ahead: the outcomes not in A form an event of their own, with probability 1 - P(A). That is the subject of complements.

Worked examples

Examples 1 to 3 go with Part 1, Examples 4 to 7 with Part 2.

Boundaries

The outcomes must be equally likely. The rule gives each outcome the same share, so it gives a wrong answer for a spinner with unequal sections, a weighted die, or letters counted in place of tiles. Sometimes a different list of outcomes is equally likely, such as the tiles in Example 2; sometimes none is. Deciding which, and what to do when none is, is the work of probability models.

The sample space must be finite and fully listed. The bottom of the fraction is a count, so no outcome may be missed or listed twice. An outcome that fits an event in two ways, like seat 33 in Example 1, is still one outcome.

In two stages, order is part of the outcome. In Example 4, “track 1 then track 4” and “track 4 then track 1” are different cells. An event described without order, such as “the two numbers add to 5”, contains both.

A count must be a whole number. Working backwards from a probability gives a count only when the arithmetic comes out whole, as Example 6 (c) shows.

A probability describes the long run, not the next few trials. \displaystyle P(\text{E}) = \frac{4}{9} does not promise 4 E tiles in every 9 draws. It is the share of draws that are E over a long run of draws, with the tile put back each time. Comparing that share with what actually happened is experimental probability.

Where this goes next

When outcomes are not equally likely, a probability model gives each its own probability, and this rule is the case where they are all the same. With no list at all, experimental probability estimates from recorded results. Tables of two stages grow into compound events, and later the multiplication rule and dependent events work from the stages without the table. A sample space too large to list is measured by counting. Successes against failures, instead of against everything, is odds; counts that become areas give geometric probability.

Exercises

Part 1: one outcome or several

For Exercises 1 to 3, the twenty-sided die of Example 5, with faces 1 to 20, is rolled once. Give each probability as a fraction in simplest form, a decimal and a percent.

1. The roll is 13.
2. The roll is a multiple of 5.
3. The roll is greater than 14.
4. A prize is taped under one seat chosen at random from seats numbered 1 to 50. Find the probability that the seat number is a perfect square, as a fraction, a decimal and a percent.
5. A ten-sided die has faces 0 to 9. Find the probability that one roll shows an odd number greater than 2, as a fraction, a decimal and a percent.

For Exercises 6 and 7, a bag holds seven letter tiles spelling PEPPERS, and one tile is drawn. Give each probability as a fraction, and as a decimal and percent rounded.

6. The tile shows P.
7. The tile shows a vowel.

Part 1: three ways

8. Each probability is given in one form. Write it in the other two. (a) 0.65 \qquad (b) \displaystyle \frac{7}{8} \qquad (c) 4\%
9. A ten-sided die has faces 0 to 9. Describe an event of one roll with probability 60\%.

Part 2: two stages

For Exercises 10 and 11, a bag holds one red, one green and one blue counter. A counter is drawn, its color noted, and it is put back; then a second counter is drawn. Make the table of outcomes first.

10. Find the probability that both counters drawn are the same color.
11. Find the probability that at least one of the two counters drawn is red.

For Exercises 12 and 13, a ten-sided die with faces 0 to 9 is rolled twice. The first roll is the tens digit and the second the ones digit of a number from 00 to 99.

12. Find the probability that the two digits are the same.
13. Find the probability that the number is less than 30.
14. A playlist has five tracks, numbered 1 to 5. On shuffle, the first track is chosen at random and the second at random from the four left. Find the probability that track 5 is one of the first two played.

Part 2: comparing events

15. The twenty-sided die with faces 1 to 20 is rolled once. Which is more likely: an odd number, or a number from 1 to 9?
16. At one concert a prize is under one of 50 seats, and Corentin’s group fills 4 of them. At another the prize is under one of 30 seats, and Elowen’s group fills 3. Who is more likely to win a prize?
17. The first bag holds 2 red and 3 blue counters. The second holds 3 red and 5 blue. From which bag is a red counter more likely?

Part 2: finding a count

18. A bag holds 60 counters, and the probability of drawing a white one is 0.45. How many white counters are there?
19. On shuffle, the probability that the first song is a slow one is 20\%, and the playlist has 8 slow songs. How many songs are on the playlist?
20. A bag holds 5 red counters and some blue ones, and the probability of drawing red is \displaystyle \frac{1}{4}. How many blue counters are there?
21. A bag holds 3 green and 9 yellow counters. How many green counters must be added to make the probability of green \displaystyle \frac{1}{2}?

Reasoning

22. The word PEPPERS uses four different letters. Explain why the probability that a tile drawn from the seven PEPPERS tiles shows P is not \displaystyle \frac{1}{4}.
23. Explain why no bag of 25 counters can give a probability of exactly 0.3 of drawing red.
24. Explain why, for an event in a sample space of equally likely outcomes, the rule always gives a number from 0 to 1, and say when it gives exactly 0 and exactly 1.
25. In the setting of Exercises 10 and 11, Agatha lists the color pairs without order: red-red, green-green, blue-blue, red-green, red-blue, green-blue. Three of these six are the same color, so she gives \displaystyle \frac{1}{2} for “both the same color”. Use the table to explain why her six pairs are not equally likely, and give the correct probability.