Video Transcript
In this video, we will learn how to
find the probabilities of the complement, intersection, and union of events. We will begin by introducing some
key probability notation and definitions together with their representation on a
Venn diagram.
The complement of an event 𝐴 in a
sample space 𝑆, denoted 𝐴 prime or 𝐴 bar, is the collection of all outcomes in 𝑆
that are not elements of the set 𝐴. In other words, 𝐴 prime is the
event that 𝐴 does not occur. The probability rule for
complements states that the probability of 𝐴 prime is equal to one minus the
probability of 𝐴. The complement of event 𝐴 can be
represented on a Venn diagram by the shaded region shown. We shade the entire region outside
the circle 𝐴.
We will now recall the definitions
for the intersection and union of two events. The intersection of events 𝐴 and
𝐵, denoted 𝐴 intersect 𝐵, is the collection of all outcomes that are elements of
both of the sets 𝐴 and 𝐵. In other words, 𝐴 intersect 𝐵 is
the event that both 𝐴 and 𝐵 occur. This can be represented on a Venn
diagram as shown. We shade the region that is in
circle 𝐴 and circle 𝐵. The union of events 𝐴 and 𝐵
denoted 𝐴 union 𝐵 is the collection of all outcomes that are elements of one or
the other of the sets 𝐴 and 𝐵 or of both of them. In other words, 𝐴 union 𝐵 is the
event that 𝐴 or 𝐵 or both 𝐴 and 𝐵 occur. We can represent the union of
events 𝐴 and 𝐵 on a Venn diagram by shading everything inside circle 𝐴 and circle
𝐵.
These two definitions of the
intersection and union of events lead us to a key rule of probability. The additive rule of probability
states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus
the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. This can also be shown using Venn
diagrams. When adding the probability of
event 𝐴 to the probability of event 𝐵, we add the intersection twice. This means that by subtracting the
probability of the intersection of events 𝐴 and 𝐵 from their sum, we obtain the
probability of the union.
The information we have seen so far
leads us to six further formulae that we can use to solve problems involving
probability on Venn diagrams. Firstly, we have the complement of
the union of events 𝐴 and 𝐵. This is equal to one minus the
probability of the union of events 𝐴 and 𝐵. On a Venn diagram, this can be
shown by shading everything outside of the union of events 𝐴 and 𝐵. In the same way, we have the
probability of the complement of the intersection of events 𝐴 and 𝐵. This is equal to one minus the
probability of the intersection of events 𝐴 and 𝐵. In this case, we shade everything
apart from the intersection of the two events. In the first diagram, we see that
the union and its complement sum to one, and in the second diagram, the intersection
and its complement sum to one.
Next, we will consider the
probability of exactly one event occurring. This leads us to two possible
scenarios: firstly, the probability of event 𝐴 occurring and event 𝐵 not occurring
and secondly, the probability of event 𝐵 occurring and event 𝐴 not occurring. These are the intersections of
event 𝐴 and the complement of event 𝐵 and vice versa. If we want only event 𝐴 to occur,
then we shade the region that is in set 𝐴 but not in set 𝐵. This is therefore equal to the
probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. The same method can be used in
order to demonstrate the probability that just event 𝐵 occurs. This is the probability of 𝐵 minus
the probability of 𝐴 intersection 𝐵.
Our last two formulae are the
complements of these two. Firstly, we consider the
probability of the union of the complement of 𝐴 and event 𝐵, and secondly, the
probability of the union of event 𝐴 and the complement of 𝐵. In the Venn diagram drawn, the
complement of 𝐴 is shaded in orange and event 𝐵 is shaded in pink. As we want the union of these, one
or other or both can occur. This means that we want the entire
shaded region. As we have just seen, the area not
shaded is the probability that exactly 𝐴 occurs, written the probability of 𝐴
intersection the complement of 𝐵. We can therefore conclude that the
probability of the union of the complement of 𝐴 and event 𝐵 is equal to one minus
this value. In the same way, we have the
probability of the union of 𝐴 and the complement of 𝐵 is equal to one minus the
probability of the intersection of the complement of 𝐴 and event 𝐵.
We will now look at a couple of
examples where we can use these definitions and formulae to solve problems involving
complements, intersections, and unions.
Denote by 𝐴 and 𝐵 two events with
probabilities the probability of 𝐴 is 0.2 and the probability of 𝐵 is 0.47. Given the probability of 𝐴
intersection 𝐵 is 0.18, find the probability of 𝐴 union 𝐵.
We begin by recalling the notation
for union and intersection in this question. We can then answer this question
using the additive rule of probability. This states that the probability of
𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the
probability of 𝐴 intersection 𝐵. This can also be shown using Venn
diagrams. Substituting in the values given,
we have the probability of 𝐴 union 𝐵 is equal to 0.2 plus 0.47 minus 0.18. This is equal to 0.49. If the probability of 𝐴 is 0.2,
the probability of 𝐵 is 0.47, and the probability of 𝐴 intersection 𝐵 is 0.18,
then the probability of 𝐴 union 𝐵 is 0.49.
We could also have worked this out
by completing each individual section of the Venn diagram. Clearing some space, we can begin
by adding the probability of the intersection of events 𝐴 and 𝐵, which is
0.18. As the probability of event 𝐴 is
0.2, we can calculate the probability of only event 𝐴 occurring by subtracting 0.18
from 0.2. This is equal to 0.02. Likewise, the probability of only
event 𝐵 occurring is equal to 0.47 minus 0.18. This is equal to 0.29. The union of the two events will
therefore be equal to the sum of 0.02, 0.18, and 0.29. This is equal to 0.49. To complete the Venn diagram, it is
important to add the probability that neither event 𝐴 nor event 𝐵 occur, in this
case, 0.51. This is one minus 0.49.
In our next example, we need to
calculate the probability that neither event 𝐴 nor event 𝐵 occur.
Suppose 𝐴 and 𝐵 are two events
with probability 𝑃 of 𝐴 equals 0.6 and 𝑃 of 𝐵 equals 0.5. Given that the probability of 𝐴
intersection 𝐵 is 0.4, what is the probability that neither of the events
occur?
In this question, we need to
calculate the probability that neither of the events occur. As shown on the Venn diagram, this
is the same as the complement of the union of events 𝐴 and 𝐵. As an event and its complement sum
to one, we can therefore calculate the probability that neither event 𝐴 nor event
𝐵 occur by subtracting the probability of 𝐴 union 𝐵 from one. In this question, we are not given
the probability of the union. However, we are given the
probability of event 𝐴, the probability of event 𝐵, and the probability of the
intersection of events 𝐴 and 𝐵. This means that we can begin by
using the additive rule of probability, which states that the probability of 𝐴
union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the
probability of 𝐴 intersection 𝐵.
Substituting in the values given,
the right-hand side becomes 0.6 plus 0.5 minus 0.4. The probability of 𝐴 union 𝐵 is
equal to 0.7. This is the area shaded in pink on
our Venn diagram. We can now find a complement of
this by subtracting 0.7 from one, which is equal to 0.3. We can therefore conclude that if
the probability of 𝐴 is 0.6, the probability of 𝐵 is 0.5, and the probability of
𝐴 intersection 𝐵 is 0.4, then the probability that neither event 𝐴 nor event 𝐵
occurs is 0.3. This is the section outside of the
circles on our Venn diagram.
In our final question, we will use
the additive rule of probability to solve a problem in context.
A group of 68 school children
completed a survey asking about their television preferences. The results show that 43 of the
children watch channel 𝐴, 26 watch channel 𝐵, and 12 watch both channels. If a child is selected randomly
from the group, what is the probability that they watch at least one of the two
channels?
Our aim in this question is to
calculate the probability that a randomly selected child watches at least one of the
two channels. This is the probability of 𝐴 union
𝐵 and can be represented on a Venn diagram, as shown. We recall that the additive rule of
probability states that the probability of 𝐴 union 𝐵 is equal to the probability
of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.
We are told in the question that 43
out of the 68 school children watch channel 𝐴. This means that the probability of
selecting a child at random that watches channel 𝐴 is 43 out of 68. Likewise, the probability of
selecting a child that watches channel 𝐵 is 26 out of 68, as the results of the
survey showed that 26 of the children watch channel 𝐵. We were also told that 12 of the
children watch both channels. This is the intersection of events
𝐴 and 𝐵 as shown in pink on the Venn diagram. The probability of this
intersection is equal to 12 out of 68.
Substituting the three fractions
into our formula, we have the probability of 𝐴 union 𝐵 is equal to 43 over 68 plus
26 over 68 minus 12 over 68. As the denominators are the same
and 43 plus 26 minus 12 equals 57, the probability of 𝐴 union 𝐵 is 57 over 68. We can therefore conclude that the
probability that a randomly selected child watches at least one of the two channels
is 57 over 68.
We will now summarize the key
points from this video. The complement of an event 𝐴,
denoted 𝐴 prime, is the collection of all outcomes that are not elements of set
𝐴. The intersection of events 𝐴 and
𝐵, denoted 𝐴 intersect 𝐵, is the collection of all outcomes that are elements of
both of the sets 𝐴 and 𝐵. The union of events 𝐴 and 𝐵 is
the collection of all outcomes that are elements of one or the other of the sets 𝐴
and 𝐵, or both of them.
These definitions lead us to the
following formulae. The additive rule of probability
states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus
the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. Other rules of probability
involving complements, unions, and intersections that we have seen in this video
include the following six. We can use these rules together
with Venn diagrams to calculate probabilities and solve problems in context.
