Use the following probabilities to answer the question.
It may be helpful to sketch a Venn diagram, which we will do.
We're given the probability of A equals zero point three, five,
the probability of B equals zero point four,
and the probability of A and B equals
zero point two, five. We want to find the probability of B
given not A, which is conditional probability
to find here below. So the probability of B
given not A, equals the probability of
not A and B,
divided by
the probability of not A.
Now to help us do this, let's complete the Venn diagram.
We'll begin with the intersection of A and B. Notice a probability of A and B equals zero point two, five.
So the probability of being in this region here is zero point two, five.
Now the probability of A equals zero point three, five,
but this does include this probability because this an A,
and therefore, the probability of being in this region which is only A,
would be zero point three, five minus zero point two, five, or zero point one.
And then because the probability of B equals zero point four,
which again includes this probability. The probability of only B
would be zero point four minus zero point two, five, or zero point one five.
We also want to find the probability
of not A and not B, or the probability of being in this outer region.
Well the probability of only being in the outer region would be
one minus the sum of these probabilities,
which would be zero point five. Again we can find this by taking one,
which is the probability of being in the universal set. And then subtracting the
sum of these three probabilities.
So point one plus point two, five,
plus point one, five, again gives us the probability of being
in the universal set, but not in A and not in B.
Now before we find these two probabilities, let's highlight
the region it would be not A and B, as well as a region that's not A.
The region that's not in A
and in B would have to be the region here that's only in B.
Again this is the region that is
not in A, and also
in B. And then the region that's not A
would be all the region that 's outside set A,
which would be this entire region here.
So the probability of not A
and B is the probability of being only in B which is zero point one five,
divided by
the probability of not A. To find that probability
we would find the sum of these two probabilities,
or one minus the sum of these two probabilities.
Notice zero point five plus zero point one, five is zero point six, five.
So notice how to find the probability
of B, given not A, we found the probability of the desired region which is only B,
and then divided by the probability of the total region
given and not A, which is the probability
zero point six, five. And now we'll apply this probability as a simplified fraction
as well as a decimal and percentage. To clear the decimals, we can multiply by
one hundred over one hundred, which gives us
fifteen sixty-fifths. But now we have a common factor of five,
so dividing numerator and denominator by five, our simplified probability as a fraction
is three-thirteenths. Let's also convert to a decimal and percentage.
To convert a decimal we divide. Three divided by thirteen,
to four decimal places would be approximately point two, three,
zero, eight.
To convert to a percentage we multiply two buy one hundred,
add a percent sign, or move the decimal to the right two places
and add a percent sign, giving us twenty-three point zero, eight
percent. Notice how the only exact probability is the one in fraction form
so for our homework, we should enter the fractional probability
even though we often do see probability as a decimal
and percentage. But here notice how they are approximations
because we did round.
I hope you found this helpful.