Question

We have learned about tree diagrams and venn diagrams. Explain what each of these are and how they are helpful when calculating probability. Then post a problem that your classmates can solve using one of these two methods and respond to one of their questions as well.

Answer 1

Tree diagrams. Tree diagrams are a way of showing combinations of two or more events. Each branch is labelled at the end with its outcome and the probability is written alongside the line. ... There are four possible outcomes( given example). To work out the probabilities of each combination, multiply the probabilities together.

Let's take a look at a simple example, flipping a coin and then rolling a die. We might want to know the probability of getting a Head and a 4.
If we wanted, we could list all the possible outcomes:

(H,1) (H,2) (H,3) (H,4) (H,5) (H,6)

(T,1) (T,2) (T,3) (T,4) (T,5) (T,6)

Probability of getting a Head and a 4:

P(H,4) = 1/12

Here is one way of representing the situation using a tree diagram. To save time, I have chosen not to list every possible die throw (1, 2, 3, 4, 5, 6) separately, so I have just listed the outcomes "4" and "not 4":

Each path represents a possible outcome, and the fractions indicate the probability of travelling along that branch. For each pair of branches the sum of the probabilities adds to 1.

A Venn diagram is a graphical way of representing the relationships between sets. In each Venn diagram a set is represented by a closed curve.

The region inside the curve represents the elements that belong to the set, while the region outside the curve represents the elements that are excluded from the set.

Venn diagrams are helpful for thinking about probability since we deal with different sets. Consider two events, \(A\) and \(B\), in a sample space \(S\). The diagram below shows the possible ways in which the event sets can overlap, represented using Venn diagrams:

The sets are represented using a rectangle for \(S\) and circles for each of \(A\) and \(B\). In the first diagram the two events overlap partially. In the second diagram the two events do not overlap at all. In the third diagram one event is fully contained in the other. Note that events will always appear inside the sample space since the sample space contains all possible outcomes of the experiment.