A survey was conducted
at a local ballroom dance studio
asking 90 students if they had ever competed
in the following dance categories: Smooth, Rhythm, Standard.
The results are shown here below in the Venn diagram,
and because 90 students were surveyed,
if we add all of these numbers, the sum is 90,
which is important to recognize because remember,
the probability of an event is equal
to the favorable number of outcomes
divided by the total number of outcomes,
and because we are selecting one student at random
from 90 students, the total number of outcomes
will always be 90 for each of the probabilities.
We first want to find the probability
that if one student is chosen at random,
the student has competed in none of the categories.
So looking at the Venn diagram,
these eight students are the students
that have not competed in any of the three categories,
and therefore, they are eight favorable outcomes
out of a total of 90,
giving us a probability of eight divided by 90, or 8/90.
Eight and 90 share a common factor of two.
To simplify, we divide both by two.
Eight divided by two is four.
90 divided by two is 45.
The simplified probability is 4/45 as a fraction,
but let's also show the probability
as a decimal and a percent.
If we want the percent to have one decimal place,
the decimal must have three decimal places.
So let's go to the calculator
and convert the fraction to a decimal.
Four divided by 45, Enter,
to three decimal places is .089, or 0.089,
which gives us 8.9%.
Just keep in mind, the fraction is exact,
and these two are rounded.
Next, you want the probability
the student has competed in all three of the categories.
We can find these students in the intersection
of the three sets, which is this set here,
because 11 students have competed in all three categories
out of the 90 students, the probability is 11/90,
or 11 divided by 90.
Again, let's also show the decimal and percentage.
So, 11 divided by 90 to three decimal places is 0.122,
which is equal to 12.2%.
Next, we want the probability
the student has competed in Smooth or Standard,
but not Rhythm.
Let's first determine the set that represents
the students that competed in Smooth or Standard,
which is going to be the union of the Smooth
and Standard sets.
So all of these students competed in Smooth or Standard,
but we're looking for the students
that competed in Smooth or Standard but not Rhythm,
which means now we must remove the part of the set
that also includes rhythm, which is this
part of the set here.
So if we remove this part of the set,
it leaves us with the set here in red,
where the students in this set in red
have competed in Smooth or Standard, but not Rhythm.
And now we need to figure out
how many students this would be.
There are 10 plus 12 plus seven, which is 29 students.
There are a total of 29 favorable outcomes
out of a total of 90 outcomes
for the student to have competed in Smooth or Standard,
but not Rhythm.
So the probability is 29/90, which doesn't simplify.
Let's also find the decimal and percentage.
29 divided by 90,
to three decimal places, is 0.322,
which is equal to 32.2%.
Next, we want the probability the student
has competed in Rhythm and Standard, but not Smooth.
Let's first find the students that competed
in Rhythm and Standard by determining the intersection
of the Rhythm and Standard sets.
So the intersection of the Rhythm and Standard sets
is this set here,
but notice how this part of the set
overlaps with the Smooth set,
so these 11 students also competed in Smooth,
and we're looking for the number of students
that competed in Rhythm and Standard but not Smooth,
so if we remove these 11 students from the set
we're left with the set here in red,
which again, these 13 students
competed in Rhythm and Standard, but not Smooth,
and therefore there are 13 favorable outcomes
out of a total of 90 outcomes.
The probability is 13/90,
which converting to a decimal
is approximately 0.144,
which gives us 14.4%.
And for the last probability,
we want the probability the student has competed in Rhythm.
So they save the easiest one for last.
The students that competed in Rhythm
will be in the Rhythm set, which is this set here,
and there are a total of 14 plus 15 plus 11 plus 13,
or 53 students in the Rhythm set,
and therefore, there are 53 favorable outcomes
out of a total of 90 outcomes.
The probability is 53/90.
Let's convert this to a decimal and percent.
53 divided by 90
is approximately 0.589,
which gives us 58.9%.
I hope you found this helpful.