Math Vale

🎲 Probability Marsh · Probability

Outcomes & Probability

Measure how likely something is on a scale from 0 to 1 by comparing favourable outcomes with all the outcomes there are.

In short

  • Every probability lies between 0 (impossible) and 1 (certain), so an answer outside that range is wrong before you check the arithmetic.
  • The sample space is the list of everything that could happen; for a two-stage experiment it is a grid, and the number of outcomes is rows times columns.
  • When outcomes are equally likely, P(event) = favourable outcomes / TOTAL outcomes — the whole, not the leftovers.
  • The same probability can be written as a fraction, a decimal or a percent, and comparing two probabilities means comparing two fractions.

A number for how likely something is

Probability turns "probably" into a number. That number always lives on a scale from 0 to 1:

  • 0 means impossible. It will not happen.
  • 1 means certain. It will happen.
  • 1/2 means an even chance — as likely as not.

Everything else lies in between. A probability of 0.2 is unlikely; 0.9 is likely. Nothing sensible ever comes out negative or above 1, so those two ends are your first and best check on an answer: if you get 3/2, something has gone wrong before you reached the arithmetic.

The same probability can be written three ways, and all three mean exactly the same thing: 1/4 = 0.25 = 25%. Use whichever the question asks for.

The sample space

Before you can measure a chance you have to know what could happen. The sample space is the list of every possible outcome.

  • One coin: heads, tails. Two outcomes.
  • One six-sided die: 1, 2, 3, 4, 5, 6. Six outcomes.
  • A spinner with 8 equal sections: eight outcomes.

For a two-stage experiment, lay the sample space out as a grid: one stage down the side, the other along the top. Toss a coin and roll a die and you get a 2 by 6 grid — 12 cells, 12 outcomes. Every cell is one outcome, so the number of outcomes is rows times columns.

That grid is worth drawing even when you are confident, because it shows something a list of totals hides: with two dice there is only one way to make 2, but six ways to make 7.

Favourable over total

When every outcome in the sample space is equally likely, the probability of an event is a comparison of two counts:

P(event) = (number of favourable outcomes) / (total number of outcomes)

A bag with 3 red and 5 blue counters has 8 counters in total, so P(red) = 3/8. Not 3/5. The bottom of the fraction is always the whole sample space, never the leftovers — 3 to 5 is the *odds*, which is a different (and older) way of speaking, and it is the single most common slip in this topic.

The words "equally likely" carry real weight. They are true for a fair die, a fair coin and a spinner with equal sections. They are not true for a spinner whose sections are different sizes, and they are not true for the totals of two dice.

Describing and comparing

Once you can compute a probability you can *compare* two of them, and that is where fraction skills earn their keep. Is 3/7 or 4/9 more likely? Cross-multiply: 3 x 9 = 27 against 4 x 7 = 28, so 4/9 is the larger — barely.

Counting the favourable outcomes alone will not do it. A spinner with 5 green sections out of 20 has more green than one with 3 out of 8, and yet green is *less* likely on it, because 5/20 = 0.25 is under 3/8 = 0.375. Probability is always a share, never a raw count.

To describe a single probability in words, place it on the scale: below a half is unlikely, exactly a half is an even chance, above a half is likely.

Worked examples

Example 1

A pouch holds 4 red, 6 blue and 2 gold tokens. One is drawn without looking. What is P(gold)?

  1. First the sample space: every token is equally likely to be drawn, and there are 4 + 6 + 2 = 12 tokens.
  2. Now the favourable outcomes: 2 of them are gold.
  3. P(gold) = favourable / total = 2/12.
  4. Cancel: 2/12 = 1/6.
  5. Sense check: 1/6 is about 0.17, comfortably between 0 and 1, and gold is the rarest colour — so a small probability is exactly what we should expect.

Example 2

20 rune cards are numbered 1 to 20 and one is turned over. What is the probability that the number is a multiple of 3?

  1. The sample space is the 20 cards, all equally likely.
  2. List the favourable outcomes: 3, 6, 9, 12, 15, 18 — that is 6 cards.
  3. P(multiple of 3) = 6/20.
  4. Cancel by 2: 6/20 = 3/10 = 0.3 = 30%.
  5. Check by estimating: about a third of the numbers should be multiples of 3, and 0.3 is just under a third — right, because 20 is not itself a multiple of 3.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

A clay jar holds 1 red and 3 white marsh stones. One is drawn without looking. What is the probability that it is white? Give a fraction in lowest terms.

Answer: 3/4

  1. Total outcomes: 1 + 3 = 4.
  2. Favourable outcomes: 3 of the marsh stones are white.
  3. P(white) = 3/4 = 3/4
  4. As a decimal that is about 0.75, which sits sensibly between 0 and 1.

Problem 2

Difficulty 3 of 5

How many different outcomes are there in total when a coin is tossed and then a six-sided die is rolled?

Answer: 12

  1. Write the sample space as a grid: one stage down the side, the other along the top.
  2. Every cell of the grid is one outcome, so the number of outcomes is rows x columns.
  3. That gives 12 outcomes altogether.

Problem 3

Difficulty 4 of 5

Yuan has 25 counters in a pouch, and 23 of them are marked with a rune. One is drawn without looking. What is the probability that it is marked? Give your answer as a percent.

Answer: 92 %

  1. P(marked) = 23/25
  2. 23 / 25 = 0.92
  3. 0.92 x 100 = 92, so the probability is 92%.
  4. Sense check: a probability always lands between 0 and 1 (or between 0% and 100%).

Common mistakes

  • Writing favourable against unfavourable (3 red to 5 blue as "3/5") instead of favourable over total, which gives odds rather than a probability.
  • Answering with the count of favourable outcomes rather than the probability — "3" instead of "3/8".
  • Treating outcomes as equally likely when they are not, such as assuming each total on two dice is as likely as any other.
  • Comparing the number of favourable outcomes on two spinners rather than the share each represents.
  • Forgetting to multiply by 100 when a probability is asked for as a percent, or dividing twice and giving 0.0025 for 25%.

What you should be able to do

  • List the sample space of a one- or two-stage experiment and count its outcomes.
  • Write the probability of an event as a fraction in lowest terms.
  • Express the same probability as a fraction, a decimal and a percent.
  • Describe an event as impossible, unlikely, even, likely or certain, and compare two events.

Where this fits in the curriculum

Common Core

  • 7.SP.C.5

    Grade 7 — Understand that the probability of a chance event is a number between 0 and 1 expressing how likely it is to occur.

  • 7.SP.C.7.A

    Grade 7 — Develop a uniform probability model by assigning equal probability to all outcomes, and use it to find probabilities of events.

  • HSS-CP.A.1

    High school — Describe events as subsets of a sample space, using characteristics of the outcomes or unions, intersections and complements of other events.

Ontario

  • G7.D2.1

    Grade 7 — Use mathematical language, including the terms "impossible", "possible" and "certain", to describe the likelihood of events happening, and use that likelihood to make predictions and informed decisions.

  • G6.D2

    Grade 6 — Describe the likelihood that events will happen, and use that information to make predictions.

    Ontario numbers its Data expectations per grade document and the specific numbering could not be verified line by line here, so this tag names the strand's OVERALL expectation rather than guessing at a specific one.

SAT

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