Answer:
# of students in neither choir nor band = 160
# of students in choir only = 50
# of students in band only = 180
# of students in both choir and band = 60
Probability = 0.64
Explanation:
First, I'll provide the steps to complete the Venn diagram. Then, I'll provide the steps to find the probability that a student chosen at random is in the choir or the band or both.
Steps to complete the Venn diagram:
Step 1: Write 60 in the middle of the Choir and Band circles:
- Since we're told that 60 students are in both choir and band, you can write 60 in the oval shape in the middle, which represents the union of choir and band.
Step 2: Find the number of students who are in choir only:
The 110 students in choir is comprised of:
- The students in both choir and band,
- and the students in choir only.
- We know that of the 110 students in choir, 60 are in both choir and band.
Thus, we can find the number of students in choir only by subtracting 60 from 110:
Students in choir only = 110 - 60
Students in choir only = 50
Thus, 50 students are in choir only so write this in the choir circle.
Step 3: Find the number of students who are in band only:
Similarly, the 240 students is comprised of:
- The students in both choir and band,
- and the students in band only.
- We know that of the 240 students in band, 60 are in both choir and band.
Thus, we can find the number of students in band by subtracting 60 from 240:
Students in band only = 240 - 60
Students in band only = 180
Thus, 180 students are in band only so write this in the band circle.
Step 4: Find the number of students in neither choir nor band:
We can find the number of students in neither choir nor band by subtracting from 450 the sum of:
- the number of students in choir only,
- the number of students in both choir and band,
- and the number of students in band only.
Number of students in neither choir nor band = 450 - (50 + 60 + 180)
Number of students in neither choir nor band = 450 - 290
Number of students in neither choir nor band = 160
Thus, 160 students are in neither choir nor band so write this in the small blue square in the top left-hand corner.
Step 5: Check the validity of answers:
We need to make sure that 450 is the sum of:
- the number of students in choir only,
- the number of students in band only,
- the number of students in both choir and band,
- and the number of students in neither choir nor band.
50 + 180 + 60 + 160 = 450
230 + 220 = 450
450 = 450
Thus, all of our answers for the Venn diagram are correct.
Steps to find the probability that a student chosen at random is in the choir or the band or both:
Let's call the total number of students in choir A and the total number of students in band B.
Step 1: Find the number of students in choir or band or both:
We can find the number of students in choir or band or both using the formula:
A + B - A ∩ B, where
- A ∩ B is the number of students in both choir and band.
Thus, we plug in 110 for A, 240 for B, and 60 for A ∩ B.
Number of students in choir or band or both = 110 + 240 + 60
Number of students in choir or band or both = 290
Thus, the number of students in choir or band or both is 290.
Step 2: Divide the number of students in choir or band or both by the total number of students:
Dividing the number of students in choir or band or both(290) by the total number of students surveyed (450) will give us the probability that a student chosen at random is in the choir or the band or both:
Probability: 290 / 450
Probability: 0.64
Thus, the probability that a student chosen at random is in choir or band or both is 0.64