a Sum all the values in the range of either the country circle or the R&B circle.
B
b Sum all the values in the range of either the intersection of country and rock circles or the intersection of R&B and rock circles.
C
c Sum all the values in the range of the R&B circle but not in the range of the rock circle.
D
d Find a value that is in the intersection of all three circles.
A
a71.3%
B
b11.3%
C
c36.2%
D
d3.8%
Practice makes perfect
a We are told that a school carried out a survey about what music the students want played at a school dance. We are given a Venn diagram that represents the results.
We are asked to find the probability a randomly chosen student wants country or R&B music to be played at the school dance.
P(countryorR&B)=?
First, we can find the numbers of students that are in range of the circles that represent either country music or R&B.
As we can see, the numbers of students in range of the country or R&B are all numbers but 76. Let's add them up.
63+8+7+10+12+89=189
We found that 189 out of 265 students want country or R&B to be played at the dance. Finally, we can calculate the probability P(countryorR&B).
P(countryorR&B)=265189≈0.713
The probability that a randomly chosen student wants country or R&B to be played at the school dance is equal to about 0.713, or 71.3%.
b This time, we will find the probability that a randomly chosen student wants rock and country music or R&B and rock music to be played at the school dance.
P(rockandcountryorR&Bandrock)=?
To calculate this probability, we will use the given Venn diagram.
First, we can highlight the students that want country and rock music to be played. These students will be in the intersection of rock and country circles.
Next, let's highlight the students that want R&B and rock music to be played.
Now, we can highlight all the students that want rock and country music or R&B and rock music to be played at the school dance.
Let's add all the colored numbers.
8+10+12=30
We found that 30 out of 265 students want rock and country music or R&B and rock music to be played on the school dance. Finally, we can calculate the probability
P(rockandcountryorR&Bandrock)=26530≈0.113
The probability is equal to about 0.113, or 11.3%.
c In this part, we will calculate the probability that a randomly chosen student wants R&B but not rock music to be played at the school dance.
P(R&Bbutnotrock)=?
To calculate this probability, we will use the given Venn diagram.
Let's highlight only the students that want R&B to be played at the dance.
Now, let's remove the students that also want rock music to be played at the dance.
We can see that 7+89=96 students want R&B but not rock music to be played at the school dance. In total, 265 students took part in the survey. With this information, we are ready to calculate the probability
P(R&Bbutnotrock)=26596≈0.362
The probability is equal to about 0.362, or 36.2%.
d Finally, we will evaluate the probability that a randomly chosen student wants all three types of music to be played at the school dance.
P(allthree)=?
Let's look at the given Venn diagram.
The students that want all three types of music are in the intersection of all the three circles — country, rock, and R&B music.
We found that only 10 out of 265 students are in the intersection of all three circles, and therefore want all three types of music to be played. We can now calculate the probability P(allthree).
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