A test was given to a group of students.
The grades and gender are summarized below.
We were asked to determine two conditional probabilities
and while we could use this formula given below
for a conditional probability,
I think for these two examples,
we'll use our reasoning skills
and the information from the table
to determine the probabilities.
So for number one, if one student is selected at random,
find the probability the student received a C
given the student is female,
and number two, if one student is selected at random,
find the probability the student is female
given the student received a C.
So again, for number one,
we want to find the probability
that the student received a C
given, we know the student is female,
so given female.
For number two, we want to find the probability
that the student is female
given, we know the student received a C.
So here we got a probability of female
given the student received a C.
So in considering these conditional probabilities,
we want to first focus on what we know is given.
So for number one, we're given that the student is female,
so looking at the table, notice how this row
gives us information about the females
and we can see from the table,
there are a total of 35 females in the class
and therefore, 35 is going to be the denominator
of the probability.
Next, of these 35 females,
to determine their probability,
the student selected received a C given they're female,
we want to know how many of these females received a C.
We'll notice how this column here
gives us information about the Cs
and notice how 12 of the 35 females received a C
which means the probability the student received a C
given we know they're female will be 12/35.
Now this fraction does not simplify
because there are no common factors between 12 and 35
and therefore, this is the probability
expressed as a fraction,
let's also express this as a decimal and percentage.
So to convert to a decimal, we would divide 12 by 35.
So 12 divided 35,
we'll have to round this
and let's round to four decimal places.
This will be approximately 0.3429.
As a percentage, this would be 34.29%.
So notice how the exact probability
is given as this fraction,
the decimal in percent are only approximations
because we did round.
Now for number two, we want to find the probability
the student selected is female
given, we know the student received a C.
So here we're given, we know the student received a C,
so going back to the table,
let's begin by determining how many students received a C
and again, we can determine the number of Cs
from this column here.
Notice how the total number of C students in the class is 32
which again will be our denominator for the probability
because we know it's given the student received a C
and then from the C students,
we want to determine how many of them are female.
So again, here's the information about the females.
Notice how 12 of the C students are female
and therefore, the probability of selecting a female
given we know the student received a C will be 12/32.
Now here, this fraction does simplify.
There's a common factor of four,
so this simplifies nicely to 3/8.
Of course we can check this on the calculator.
To do this, we would enter 12 divided by 32, enter,
and if we press math, enter, enter,
it will give us the fraction back in a simplified form,
verifying 3/8 is a simplified probability as a fraction.
Notice how as a decimal
we have a terminating decimal of 0.375
which would be equal to 37.5%.
So notice in this case, the fraction, decimal, and percent,
are all exact probabilities
because we didn't have to do any rounding.
I hope you found this helpful.