Video Transcript
The probability that event 𝐴
occurs is three-fifths. If event 𝐴 does not occur, then
the probability of event 𝐵 occurring is two-thirds. What is the probability that event
𝐴 does not occur and event 𝐵 occurs?
We’re given two pieces of
information in this question. The first is that the probability
of event 𝐴 occurring is three-fifths. This is straightforward, and we can
write this 𝑃 of 𝐴 is equal to three over five. The second piece of information is
a conditional statement. This says that if event 𝐴 does not
occur, then the probability of event 𝐵 occurring is two-thirds. And we write this as the
probability of 𝐵 given not 𝐴 is equal to two-thirds. Remember, not 𝐴 is the complement
of 𝐴 and this is written as 𝐴 with a bar on the top. And remember also that since the
sum of all probabilities must equal one and that either 𝐴 or not 𝐴 must occur,
then the probability of not 𝐴 is one minus the probability of 𝐴. And this will come in useful later
on.
So we have the probability of 𝐴 is
three-fifths and the probability of 𝐵 given not 𝐴 is two-thirds. The vertical bar in this expression
reads given not, which is the conditional. We’re asked to find the probability
of not 𝐴 and 𝐵. That’s the probability that event
𝐴 does not occur and event 𝐵 does occur. And this symbol that looks like an
n represents “and” or the intersection of two events. So we have a simple probability, a
conditional probability, and we’d like to find an intersection. And we can use the formula for
conditional probability to find our intersection. This formula tells us that for two
events 𝐶 and 𝐷, the probability of 𝐶 given that 𝐷 has occurred is the
probability of 𝐶 and 𝐷 divided by the probability of 𝐷.
Now, if we let 𝐶 in the formula
correspond to our event 𝐵 and 𝐷 in the formula is our event not 𝐴, then we can
rewrite the formula as the probability of 𝐵 given not 𝐴 is equal to the
probability of 𝐵 and not 𝐴 over the probability of not 𝐴. In our numerator, we know that the
probability of 𝐵 intersection not 𝐴 is the same as the probability of not 𝐴
intersection 𝐵. So we can switch our 𝐵 and our not
𝐴 in the probability in the numerator.
And we see now that the numerator
is actually what we’re looking for. That’s the probability of not 𝐴
intersection 𝐵. And we can make this the subject of
our equation by multiplying through by the probability of not 𝐴. We can cancel this through on our
right-hand side because the probability of not 𝐴 divided by the probability of not
𝐴 is equal to one so that we have the probability of not 𝐴 times the probability
of 𝐵 given not 𝐴 is equal to the probability of not 𝐴 intersection 𝐵.
Now we know that the probability of
𝐵 given not 𝐴 is two-thirds. We also know that the probability
of 𝐴 is three-fifths. And remembering that the
probability of not 𝐴 is one minus the probability of 𝐴, we have the probability of
not 𝐴 is one minus three-fifths, which is two-fifths. So in our formula, we have that the
probability of not 𝐴 intersection 𝐵 is two-fifths times two-thirds. Multiplying fractions, we simply
multiply the numerators and the denominators so that our numerator’s two times two
is equal to four and our denominator is five times three, which is 15, so that the
probability that event 𝐴 does not occur and event 𝐵 occurs is four out of 15.
