Video Transcript
The figure shows a Venn diagram
with some of the probabilities given for two events π΄ and π΅. Work out the probability of π΄
intersection the complement of π΅. Work out the probability of π΄. Work out the probability of π΅
given π΄.
Letβs begin by having a quick look
at the Venn diagram and seeing what some of these values represent. We have two events labeled π΄ and
π΅. And in the center, weβve got this
value of 0.2, which represents the probability of the event π΄ intersection π΅. In other words, itβs the
probability of event π΄ and π΅ occurring. The value of 0.4 represents an
event being π΅ but not π΄. Notice that outside of the circles
for π΄ and π΅, we have this value of 0.3. This represents the probability of
neither π΄ nor π΅ occurring. So letβs take a look at the first
question to work out the probability of π΄ intersection the complement of π΅. This bar represents the complement,
and it really means not π΅.
When weβre finding the probability
of π΄ intersection the complement of π΅, itβs equivalent to finding the probability
that π΄ occurs but π΅ does not. It would look like this on the Venn
diagram. But how exactly do we calculate a
probability? Well, weβll need to remember a very
important fact about probabilities. The sum of the probabilities of all
outcomes must equal one. In other words, something has to
happen. Whether the outcome is event π΄ or
π΅ or both or neither, all of these probabilities will add to give us one. In fact, the only probability
thatβs missing on our diagram is that of the probability of π΄ intersection not
π΅. Therefore, if we add together 0.2,
0.4, and 0.3 and subtract these from one, then weβll find our missing
probability. So the probability of π΄
intersection the complement of π΅ is 0.1.
Letβs have a look at the second
question to work out the probability of π΄. When it comes to answering a
question like this, it can be very tempting just to give the value of 0.1. However, remember that 0.1
represents event π΄ and not π΅, and so we also need to include those events which
are π΄ and π΅, in other words, everything thatβs in the circle for event π΄. Adding the probabilities in of 0.1
and 0.2 gives us 0.3. And so the answer for the second
part of this question is that the probability of π΄ is equal to 0.3.
In the final part of this question,
weβre asked to work out a conditional probability, the probability of π΅ given
π΄. We should remember the formula for
conditional probability that the probability of π΄ given π΅ is equal to the
probability of π΄ intersection π΅ divided by the probability of π΅. Before we rush to apply this
formula, however, thereβs a change that weβll need to make. This formula allows us to calculate
the probability of π΄ given π΅, but we actually need to work out the probability of
π΅ given π΄. Weβll, therefore, need to switch
our values of π΄ and π΅ in the formula.
Notice that when weβve done so, the
numerator of our fraction is the probability of π΅ intersection π΄. The probability of π΅ intersection
π΄ is actually the same as the probability of π΄ intersection π΅. It still refers to this portion
which is both event π΄ and event π΅. Letβs now plug in the values to
find our conditional probability. The section thatβs highlighted is
the probability of π΅ intersection π΄, and remember that we calculated in our second
question that the probability of π΄ is 0.3. We can simplify this fraction by
multiplying the numerator and denominator by 10. So the answer for the third part of
the question is that the probability of π΅ given π΄ is two-thirds.
We can think about this visually on
the Venn diagram by remembering that itβs given that we have event π΄, then whatβs
the probability that itβs also event π΅. So it have the proportion of 0.2
out of 0.3, which is the fraction two-thirds.