A survey was conducted at a local ballroom dance studio
asking 67 students if they had ever competed in the
following dance categories; smooth, rhythm, standard.
Use the Venn diagram to determine the probabilities.
Express the probabilities as percentages rounded
to the tenth.
Because they want the probabilities as percents to
the tenths, we'll have to first get the probabilities
as a decimal to the thousandth or three decimal places.
Looking at the Venn diagram notice how the number
of students in each set are indicated by the numbers in
the Venn diagram, so if we added up all these numbers,
including these four students here, the sum would be
67 students.
We're first asked to find the probability a student
competed in smooth.
Looking at the Venn diagram here is the smooth set,
which indicates the number of students that competed
in smooth.
The number of students is 12 plus six plus nine plus 14,
which is 41.
41 out of 67 students competed in smooth and therefore
the probability of smooth is 41 67ths and now let's
get the decimal in percent.
41 divided by 67 to three decimal places is approximately
zero point six one two.
That is equal to 61.2%.
Next we're asked to find the probability a student
competed in rhythm and standard.
We find the students that competed in rhythm and standard
in the inner section of the rhythm and standard sets.
So looking at the Venn diagram, here we have the rhythm
set, here we have the standard set.
The inner section of the two sets is this set here.
Notice nine plus 10 or 19 out of 67 students competed
in rhythm and standard.
The probability of rhythm and standard is 19 67ths.
Which to three decimal places is approximately zero point
two eight four.
Which is 28.4%.
Next we have the probability of smooth or standard.
We find the students that competed in a smooth or standard
in the union of the smooth and standard sets.
So here's the smooth set, here's the standard set.
The union of the two sets is this set here.
The number of students that competed in smooth or standard
is 12 plus six plus nine plus 14, which we know is 41
from the fist probability, and then we have plus
10 plus five, which gives us a total of 56 out of 67
students competing in smooth or standard and therefore
the probability of smooth or standard is 56 divided
by 67 or 56 67ths.
To three decimal places we have approximately zero point
eight three six.
Which is 83.6%.
And then for the last example we're asked to find
the probability of not smooth.
So which are all the students that are outside the smooth
set or not in the smooth set.
So if this is the smooth set,
all the students not competing in smooth are in this
set here.
Well we know there are 67 students altogether and we know
there are 41 students that competed in smooth.
67 minus 41 is equal to 26.
There are 26 students not in the smooth set,
or it can also determine the 26 by adding seven, 10,
five and four, which is 26.
26 out of 67 students did not compete in smooth and
therefore the probability of not smooth is 26 67ths.
Which is approximately zero point three eight eight.
Which is 38.8%.
I hope you found this helpful.