Video Transcript
In an animal rescue shelter, 42
percent of the current inhabitants are cats, and 38 percent are dogs. Find the probability that an animal
chosen at random is not a cat. Then find the probability that an
animal chosen at random is neither a cat nor a dog.
So the first part of this problem
is taking a look at cats because what we want to do is find the probability that an
animal chosen at random is not a cat. So first of all, we are told that
42 percent of the current inhabitants are cats. So therefore, the probability of it
being a cat that is chosen is 0.42. And we get that because 42 percent
means 42 out of 100. So if we divide 42 by 100, this is
equal to 0.42. And that’s because if we divide by
100, we move each of our digits two place values to the right.
Well, in this problem, what we’re
actually looking for is the probability that an animal chosen at random is not a
cat. So what we have to help us with
this is a rule or formula. And that is that the probability of
an event not occurring or the complement of an event, as it’s also known, is equal
to one minus the probability of that event occurring. And it’s also worth reminding
ourselves at this point that we’ve got here 𝑃, and then we’ve got inside our
parentheses 𝐴 with a horizontal line above it. And we could also write this as 𝑃
and then in parentheses 𝐴 with a prime. And this would give us the same
thing because it’s just different notation to mean the complement of 𝐴 or the
probability of the complement of 𝐴.
So if we apply this rule to our
problem, then we’re gonna have that the probability that the animal chosen at random
is not a cat is equal to one minus 0.42, which is equal to 0.58. So therefore, we can say that this
is the probability that an animal chosen at random is not a cat.
Okay, so now let’s move on to the
second part of the question. Well, for part two, we already know
that the probability of a cat being chosen is equal to 0.42. And therefore, we’ve also got the
probability of a dog being chosen is equal to 0.38. And we get that because it’s 38
percent of the inhabitants are dogs. So 38 divided by 100 gives us our
0.38.
Now, for the second part of the
problem, we could just solve it using a rule we know. But it is always useful to have a
look at what different notation can be used. So we’re looking at the probability
that an animal chosen at random is neither a cat nor a dog. We can write this as we have done
here because we’ve got this U shape in between our complements. And that means all or the
union.
Well, we know that all of the
probabilities of any of the outcomes that can occur for a certain event must add up
to one. So therefore, we can say that the
probability that an animal chosen at random is neither a cat nor a dog is gonna be
equal to one minus and then if we add together the probability of a cat being chosen
and the probability of a dog being chosen. So this is gonna be one minus 0.42
plus 0.38, which is gonna give us the probability that an animal chosen at random is
neither a cat nor a dog of 0.2.
