Download Using Venn Diagrams to Solve Probability Problems

Using Venn Diagrams to Solve Probability Problems
Sometimes probability is illustrated in a Venn Diagram.
Take a look at the picture below of a Venn Diagram:
Every Venn Diagram is made up of three parts:
1. The event(s). The colored “blobs” that are inside the diagram represent
the events of the probability problem
2. The intersection(s). This is not illustrated in the figure above. But, often
there are common outcomes among events and to illustrate them in a
Venn Diagram means to have them overlap
3. The Sample Space. This is the rectangular area that surrounds all the
events. This represents the “extra” probabilities that may not be part of the
events or their intersections, but are still part of the problem to be solved.
In the case of the events taking up all the probabilities, a rectangular box
around the “blobs” is not drawn
This is an example of another Venn Diagram. Notice that events A and B share
some common outcomes. Therefore they will overlap.
Let’s solve a problem using a Venn Diagram:
Deborah and Matthew are anxiously awaiting word on whether they have
been made partners in their law firm. Deborah guesses that the she has
about a 70% chance of making partner. She thinks Matt’s chances are
about even (i.e., 50%). She also believes that there is a 30% chance that
both of them will be promoted together. With that information answer the
following questions:
1. What is the probability that Deborah is promoted and Matt is not?
2. What is the probability that Matt is promoted and Deborah is not?
3. What is the probability that neither of them is promoted?
Let’s begin by drawing a Venn Diagram and then answering each question:
Notice that each event (A = Deborah promotion and B = Matt promotion) and
their intersection is included. The part that represents A includes everything that
is related to Deborah’s promotion. The same is true for Events B and Matt’s
promotion. The size of the events doesn’t matter as long as you label them
Now, let’s look at answering each question
1. What is the probability that Deborah is promoted and Matt is not?
The answer to this is not 0.7. You may think so but that is what Deborah
guesses and does not take into consideration that she and Matt are
promoted together. In the diagram below, the actual probability is covered
in red. This part of the graph represents the intersection of everything that
is “Deborah only” and everything that is “not Matt”. That is why we use the
symbol “A
”. The actual answer to this is 0.4. This is obtained by
knowing that all of event A (Deborah promotion) sums to 0.7. But notice
that part of event A is 0.3 and this represents a shared outcome with Matt.
So, that part gets subtracted out …. 0.7 – 0.3 = 0.4
2. What is the probability that Matt is promoted and Deborah is not?
Like in the first problem, then answer is not what you may think (0.5). In the
diagram below, the actual probability is covered in blue. This part of the
graph represents the intersection of everything that is “Matt only” and
everything that is “not Deborah”. That is why we use the symbol “B A ”.
Can you see now that this problem is approached in the same manner as
problem #1? The actual answer to this is 0.2 (0.5 – 0.3 = 0.2)
So, we can actually redraw our Venn Diagram with more precise probabilities. Do
you see that the three numbers only add to 0.9? That means that 0.1 is still left
over and that fact can be used in the next problem
3. What is the probability that neither of them is promoted?
Notice that the sample space represents everything that is “not Deborah”
and everything that is “not Matt”. So, if neither of them is promoted then
neither Deborah nor Matt is promoted. Since the other probabilities include
some form or Deborah or Matt or both being promoted it should be clear
that the sample space must represent all the other possibilities