4.6 Venn diagrams, tree diagrams and combined events

Most questions about chance involve more than one event: a student who studies both physics and chemistry, two cards dealt from a pack, a test that is passed at the first or second attempt. Probabilities of such combined events can be found by organising the outcomes in a diagram and then applying a small number of rules for "and", "or" and "not".

Four diagrams do this work: the Venn diagram, the sample space diagram, the two-way table and the tree diagram. Each suits a different kind of problem, and choosing the right one is often the most important step in a solution.

Combining events

Intersection, union and complement

Two events $A$ and $B$ from the same sample space can be combined in several ways. The words "and", "or" and "not" in a question correspond to three pieces of notation borrowed from set theory.

In probability, "or" always includes the possibility of both, unless a question says otherwise (for example "$A$ or $B$ but not both"). The union $A \cup B$ therefore contains the intersection $A \cap B$. The complement was introduced in 4.5, where complementary events were shown to satisfy $P(A) + P(A') = 1$

Venn diagrams

Regions of a Venn diagram

A Venn diagram for two events that can occur together divides the sample space into four regions. Every outcome lies in exactly one of them, so the four probabilities add up to $1$.

Any combination of $A$, $B$ and their complements is a union of some of these four regions. Shading the region for an expression is a reliable way to understand it, and a common short question.

Venn diagrams with frequencies

When a question gives the numbers of people or items in each group, write the numbers in the regions. Always start with the overlap, then subtract to find the other regions, and finally use the total to find the number outside the circles.

Venn diagrams with probabilities

The same method works with probabilities in place of numbers. The total of all regions is then $1$ instead of the number of people.

Venn diagrams for three events

Three events give three overlapping circles and eight regions. They are filled in the same order: the centre (all three) first, then the regions where exactly two circles overlap, then the parts of each circle that belong to it alone, and finally the outside.

The addition rule

Deriving the rule

Adding $P(A)$ and $P(B)$ does not give $P(A \cup B)$ in general, because the overlap $A \cap B$

With equally likely outcomes, $n(A \cup B) = n(A) + n(B) - n(A \cap B)$, and dividing by $n(U)$

The rule also shows that two events with large probabilities must overlap. If $P(A) + P(B) > 1$, then $P(A \cap B) = P(A) + P(B) - P(A \cup B) \ge P(A) + P(B) - 1 > 0$

Mutually exclusive events

Events that cannot happen together

Some pairs of events can never occur on the same trial. A single roll of a die cannot be both a $2$ and a $5$; a randomly chosen card cannot be both a heart and a club.

On a Venn diagram, mutually exclusive events are drawn as circles that do not overlap.

Sample space diagrams and tables of outcomes

Two trials in a grid

When two trials are combined, such as rolling two dice, every outcome is a pair. A sample space diagram lists all of the pairs in a grid, with the outcomes of one trial along the top and the other down the side. If each trial has equally likely outcomes, every cell of the grid is equally likely.

For two fair dice the grid has $6 \times 6 = 36$ equally likely cells. Each cell holds the total of the two scores.

Grids are not limited to dice or to totals. Any two trials with a small number of outcomes can be combined, and each cell can hold a product, a difference, the larger score, or the pair itself.

Two-way tables

Data collected about two characteristics at once, such as the year group of each student and how they travel to school, is summarised in a two-way table. Row and column totals make it easy to read off the probabilities of single events, of combined events and of events restricted to one group.

A school records how $200$ students in Years $12$ and $13$ travel to school.

Part (d) is a conditional probability: the denominator is no longer the whole group of $200$ but only the group named after "given that". Probabilities of this kind can be read from any of the diagrams, and are defined formally in 4.11.

Tree diagrams

Building a tree diagram

A tree diagram shows a sequence of trials, one stage after another. Each stage has a set of branches, one for each outcome, labelled with its probability. A path from the start to the end of the tree represents one combined outcome.

A bag contains $4$ red and $6$ blue counters. A counter is taken at random, its colour is noted and it is put back. A second counter is then taken. Because the first counter is replaced, the bag is the same for the second draw, and the probabilities on the second set of branches are the same as on the first.

From the tree:

The rule "multiply along a path" works because each branch after the first gives the probability of that outcome in the situation reached so far. The rule "add between paths" works because different paths are mutually exclusive: a single run of the experiment follows exactly one path.

With and without replacement

When the first counter is not put back, the second draw is from a bag with one fewer counter, and the probabilities on the second branches depend on what happened first. This is sampling without replacement. Dealing cards, choosing people for a committee and selecting items for quality testing are all done without replacement.

Comparing the two bags shows the effect of not replacing: the probability of two reds falls from $0.16$ with replacement to $\frac{2}{15} \approx 0.133$ without, because removing a red counter makes a second red less likely.

Trees with branches that stop

A tree does not have to be symmetrical. When an experiment stops as soon as a certain outcome occurs, only the other branches continue. Problems about repeated attempts, such as "keeps trying until successful, up to three times", produce trees of this shape.

Independent events

Events that do not affect each other

Two events are independent when the occurrence of one does not change the probability of the other. Tossing a coin and rolling a die are independent; so are the two draws in a bag problem with replacement. The two draws without replacement are not independent, because the result of the first draw changes the contents of the bag.

For independent events, the probability that both occur is found by multiplying. This is the rule already used on every tree diagram with replacement.

The rule can be checked with a sample space. A fair coin is tossed and a fair die is rolled. There are $2 \times 6 = 12$ equally likely outcomes, and only one of them is (heads, $6$). So $P(\text{H} \cap 6) = \frac{1}{12}$

Conditional probability from diagrams

A probability "given that" an event has occurred is found by restricting attention to the outcomes in that event. In a diagram or table, this means using the size of the given group as the denominator instead of the whole sample space.

For the Physics and Chemistry Venn diagram, the probability that a student studies Physics, given that the student studies Chemistry, uses only the $15$ Chemistry students, of whom $7$ also study Physics:

Written with probabilities, this is $\frac{7/40}{15/40} = \frac{P(P \cap C)}{P(C)}$

The formal treatment of conditional probability, its use for testing independence, and reversing conditions on a tree diagram are in 4.11.

On a tree diagram without replacement, the second-stage branches are already conditional probabilities: the branch labelled $\frac{3}{9}$ after a red counter is $P(\text{second red} \mid \text{first red})$.

Exam-style questions on combined events

Questions on combined events often begin with a diagram to complete and then ask for several probabilities from it. Draw the diagram neatly and label it fully, because it is usually worth marks of its own and every later part depends on it.

Tree diagrams without replacement often appear with an unknown number of items. A "show that" part then asks for an equation, and the solving part follows from it.

Choosing the right diagram is often the key to a combined events question. A Venn diagram suits events that overlap within a single trial; a sample space diagram suits two trials with a small number of equally likely outcomes; a two-way table suits data classified in two ways; and a tree diagram suits trials that happen one after another, particularly when the probabilities change.

Self-check

• To do:

  I can use the notation $A \cap B$, $A \cup B$, $A'$ and shade the matching regions of a Venn diagram.

• To do:

  I can complete a Venn diagram for two or three events from frequencies or probabilities, starting from the overlap.

• To do: