Sample Spaces and Venn Diagrams

1 · Outcomes, events and sample spaces

Flip a coin. What could happen? It could land heads or tails. That's it. Each possible result is called an outcome. Doing the thing once (one flip, one roll, one spin) is called a trial.

The list of every possible outcome is called the sample space. We often write it inside curly brackets: for a coin it is { H, T }, and for an ordinary six-sided dice it is { 1, 2, 3, 4, 5, 6 }.

An event is the thing you're interested in: "roll an even number", for example. An event is made of one or more outcomes; "even" is the outcomes 2, 4 and 6. When all the outcomes are equally likely (a fair coin, a fair dice), the probability of an event is

P(event) = outcomes in the event ÷ all outcomes

You met this on the probability scale page. The whole point of a sample space is to get the bottom number right. Pick an experiment below, then tap outcomes (or use a ready-made event) to see the probability.

2 · Listing outcomes for two things at once

Now flip a coin and roll a dice. An outcome is now a pair, like (H, 3): heads on the coin and 3 on the dice. How many pairs are there? If you write them down in a random order you'll almost certainly miss some or write some twice. The trick is to be systematic (work in a fixed order):

1. Fix the first thing at its first outcome (H).
2. Run through every outcome of the second thing: (H, 1), (H, 2), …, (H, 6).
3. Move the first thing on (T) and run through the second thing again.

Choose two items and press Next outcome to build the list one step at a time. Each row of the grid is one outcome of the first item.

Notice the pattern: a coin has 2 outcomes and a dice has 6, and the grid has 2 rows of 6, so there are 2 × 6 = 12 outcomes. In general, multiply the numbers of outcomes (don't add them). Try two coins: you get HH, HT, TH and TT. HT and TH are different outcomes, because "first coin heads, second tails" is not the same as the other way round. Imagine one coin is a 10p and the other a 2p and it's obvious.

3 · Sample space diagrams: two dice

Roll two dice. Listing 36 pairs is slow, so we draw a sample space diagram: a grid with Dice 1 down the side and Dice 2 along the top. Each of the 36 cells is one equally likely outcome. In each cell we write whatever we care about: the sum (add), the difference (bigger take away smaller) or the product (multiply).

To find a probability, count the cells in your event and put that number over 36. Tap any cell to highlight every cell with the same value, or build an event with the menus.

Look at the sums: 7 appears 6 times (a whole diagonal) but 2 and 12 appear only once each. That's why 7 is the most common total with two dice. There are 11 different totals, but they are not equally likely, so P(total is 7) is not 1/11. Always count the 36 cells.

4 · Two-way tables

A two-way table sorts a group of people (or things) in two different ways at once: one way across the rows, the other down the columns. Here, a group of students is split by boys and girls (rows) and by how they get to school (columns). The Total row and column add things up:

• each row adds across to its row total;
• each column adds down to its column total;
• the bottom-right corner is the grand total: everyone.

So if one number in a row or column is missing and you know all the others, you can always find it by adding or subtracting. Fill in the gaps below and press Check. If you're stuck, the feedback points you to a row or column with just one gap.

When a question says "a student is picked at random from the group", the bottom of the fraction is the grand total. When it says "a girl is picked at random", you are only choosing from the girls, so the bottom is the girls' total. Read who is being picked! (If the table came from a survey, these probabilities are really estimates: see theoretical and experimental probability.)

5 · Venn diagrams: ξ, ∩, ∪ and ′

A set is a collection of things, called its elements, written in curly brackets. For example, if A is "even numbers up to 10" then A = {2, 4, 6, 8, 10}. We write n(A) for how many elements are in A, so n(A) = 5.

A Venn diagram draws sets as circles inside a rectangle. The rectangle is the universal set, written ξ (the Greek letter "xi", said "ksy"): everything we are talking about. Here ξ = {1, 2, 3, …, 20}. Each number goes in exactly one region:

A memory trick: ∩ looks like a bridge between the circles ("and"); ∪ is a cup that holds everything ("or"). Drag each number into the region where it belongs. If you put one in the wrong place, you'll be told why. Then tap a button (or a region) to shade it and see its probability if one of the 20 numbers is picked at random.

6 · Filling in a Venn diagram from numbers

Often you aren't given the elements, just how many there are, like this: "30 students: 18 play football, 12 play tennis, 5 play both." Now each region holds a frequency (a count). The golden rule is start in the middle:

1. Put the "both" number in the overlap.
2. The 18 football players include the 5 who play both, so football only = 18 − 5 = 13.
3. Tennis only = 12 − 5 = 7.
4. Outside both = 30 − (13 + 5 + 7) = 30 − 25 = 5. Don't forget this region!

Your turn. Fill in all four regions and press Check, or press Show me step by step to watch it done.

Once the diagram is full, probabilities are easy: add up the regions you want and divide by the total in ξ. For "or" (∪) add all three regions inside the circles once each; for "not" (′) remember the region outside both circles.

7 · Spot the mistake

Each of these students has made one of the classic mistakes. Read their work and pick what went wrong.

Worked examples

Quick check

What you've learned

• An outcome is one possible result; the sample space is the list of all of them; an event is a group of outcomes.
• With equally likely outcomes, P(event) = outcomes in the event ÷ total outcomes.
• List two events systematically: fix the first, run through the second. The number of outcomes is the product, e.g. coin and dice: 2 × 6 = 12.
• For two dice use a 6 × 6 sample space diagram (sum, difference or product in each cell) and count cells out of 36. (2, 5) and (5, 2) are different cells.
• In a two-way table rows add to row totals and columns add to column totals; fill gaps using a row or column with only one gap. The bottom of a probability is the total of the group being picked from.
• In a Venn diagram, ξ is everything; A ∩ B is "and" (the overlap), A ∪ B is "or" (anywhere in the circles), A′ is "not A" (including outside both circles).
• To fill a Venn diagram from counts, start in the middle, subtract it from each circle total, then find the outside from the total.
• Watch for missing or repeated outcomes, counting the overlap twice and forgetting the outside region.