A test was given to a group of students. The grades and gender are summarized
below on the table.
If one student is chosen at random from those who took the test,
find the probability that the student was female,
given the student got an A. This is called conditional probability,
where the conditional probability is the probability of an event
E sub two occuring, given that an event that E sub one
has already occurred. And we do have this formula here for our conditional
probability.
But let's see if we can determine this conditional probability just by
analyzing a table first.
And then we'll verify the results using our conditional probability formula.
So we want to find the probability
that a selected student would be female
given they received an A on the test.
If we take a look at our table here,
because we know the student already received an A, we can find all the information we
need from this first column.
Notice how there were a total
of sixteen A students, and of those A student's,
nine of them were female. So if we already know the student received an A,
the probability of selecting a female student
given the student received an A, would be equal to
nine sixteenths. The total number of outcomes
is sixteen because we know they already received an A.
The favorable number of outcomes is nine, the number of females
that received an A. This is our conditional probability.
Let's also write this is a decimal and a percentage.
Nine divided by sixteen
is equal to zero point five, six, two, five,
which as a percent would be fifty-six
point two, five percent. Now let's verify this probability
using our conditional probability formula. We would have the probability
of a female, given
an A grade, is equal to this fraction here.
Where we have the probability
of E sub two and E sub one.
Where the probability of a female, and an A,
divided by the probability of E sub one
which in our case would be the probability of receiving an A.
So going back to our table,
the probability of selecting one student, and that student being
female, and receiving A, notice how there's a total of nine
female students that received an A out of the total class
of seventy-five. So the probability of selecting one student
and that student being female, and receiving an A,
would be nine, seventy-fifths,
divided by, selecting one student from the class
and the probability of that student receiving an A, notice how there were a total
of sixteen students that received an A, out of the
class of seventy-five students. And therefore the probability
of selecting a student that received an A would be sixteen
seventy-fifths. Remember this fraction bar means division.
So we can write this as nine, seventy-fifths
divided by sixteen,
seventy-fifths, which is equal to
nine, seventy-fifths times the reciprocal of
seventy-five, sixteenths. And notice how
the seventy-fives simply to one, leaving us with the probability that we found above
as nine-sixteenths.
Because we have this nice table that sbreak down all the information
for us,
we can find this conditional probability straight from the table
and not have to use this formula. But sometimes this formula is very helpful.