Question 1182107: Tonya was at the beach one sunny afternoon. There were 68 people on her section of the beach. Tonya noted whether the beach-goers wore any of the following: sunglasses, a hat, swimsuit. She noted the following:
• 56 people wore a swimsuit or sunglasses 12 people wore all 3 items
4 people were not wearing any of these items
15 people were wearing sunglasses and a hat 14 people were only wearing sunglasses
18 people were wearing exactly 2 of the 3 items
38 people were wearing sunglasses
show work!!!!
1. Draw a well-labeled Venn diagram using the above information.
2. How many people were only wearing a swimsuit?
3. What is the probability that a beach-goer was wearing a swimsuit and a hat?
4. What is the probability that a beach-goer was wearing a hat or sunglasses?
5. Among those wearing swimsuit, what is the probability that they were wearing
sunglasses?
6. Given that a beach-goer was not wearing a hat, what is the probability that they were wearing a swimsuit?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Given Info:- Fact 1: 56 people wore a swimsuit or sunglasses
- Fact 2: 12 people wore all 3 items
- Fact 3: 4 people were not wearing any of these items
- Fact 4: 15 people were wearing sunglasses and a hat
- Fact 5: 14 people were only wearing sunglasses
- Fact 6: 18 people were wearing exactly 2 of the 3 items
- Fact 7: 38 people were wearing sunglasses
- Fact 8: There were 68 people on her section of the beach
Draw a rectangle and place three circles inside it. Make the circles overlap as shown below
The circle labels are
S = sunglasses
H = hat
B = bathing suit
Fact 3 allows us to place the value 4 outside all of the circles, but still within the universal set rectangle.
From Fact 2, we can place 12 in the region between all three circles.
Now focus on Fact 4 and Fact 2. This allows us to say 15-12 = 3 people wore sunglasses and a hat only. So we'll have this value in circles S and H, but outside circle B.
Fact 5 says that we'll put the number 14 inside circle S, but outside the other circles.
Add up everything so far in circle S
14+3+12 = 17+12 = 29
So far, we count 29 people who wear sunglasses. Then we have Fact 7 say 38 people wear sunglasses.
The missing value for circle S is therefore 38-29 = 9. This value goes in between circle S and B, but outside circle H.
At this point, circle S is completely filled out.
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Now move onto Fact 6. We know that 18 people wore exactly two items. The numbers 9 and 3 represent people counted so far who wear two items
9+3 = 12
The missing value is 18-12 = 6, which goes in the overlapped region between circles B and H, but outside circle S.
Last but not least, we finally arrive at Fact 1 which says 56 people wore a swimsuit or sunglasses.
Add up everything in circles S and B:
14+3+9+12+6 = 44
The value 56-44 = 12 must be the missing item that goes in the last region of circle B
So we have 12 people who wear a bathing suit but no sunglasses and no hat either.
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We're almost done. Add up every number written in the Venn diagram so far. It doesn't matter what circle we're in, or if we're outside the circles.
So far, we count
14+3+9+12+6+12+4 = 60 people
Fact 8 says Tonya observed 68 people in total. That must mean 68-60 = 8 people go inside circle H but outside the other circles.
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There are a lot of steps in the last section above. But after you get used to the process, its not so bad.
Now onto the questions your teacher asked.
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Question 1.
The venn diagram is below
The steps are mentioned in the previous section of course.
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Question 2.
There are 12 people wearing only a bathing suit. This value is in circle B, but outside the other circles.
In other words, there are 12 people wearing a bathing suit, but they didn't wear sunglasses and they didn't wear a hat either.
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Question 3.
We have 12+6 = 18 people wearing a swimsuit and a hat (and perhaps sunglasses as well). We add up the numbers inside circles B and H.
This is out of 68 peopple total
We get a probability of 18/68 = 9/34
Note: 9/34 = 0.2647 approximately
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Question 4.
Add up all the numbers that are in circles S or H, or both regions.
14+3+8+9+12+6 = 52
Or you could note that 8 people wear a swimsuit but neither of the other items, and 4 wear none of the other items. So 12+4 = 16 people do not have sunglasses and they don't have a hat either. Therefore, 68-16 = 52 people are what we want to focus on here.
This is out of 68 total
52/68 = 26/34 is the answer in fraction form.
In decimal form, that's roughly 26/34 = 0.7647
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Question 5
We only focus on the numbers in the bathing suit circle B.
There are 9+12+6+12 = 39 people wearing a bathing suit. Among this group, we have 9+12 = 21 people wearing sunglasses.
21/39 = 7/13 is the fractional answer
The decimal form of that is 7/13 = 0.5385 approximately.
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Question 6
We have 3+8+12+6 = 29 people wearing a hat and 68-29 = 39 people not wearing a hat.
Another way to see this is to add up everything that isn't in circle H (14+9+12+4 = 39).
The universal set reduces from everyone on the beach to just non-hat people.
Of the total 39 people not wearing a hat, 9+12 = 21 people are wearing a bathing suit. Note how I'm adding the values in circle B but outside circle H.
Divide the values:
21/39 = 7/13
We get the same answer as before. It's likely a coincidence, but I'm not entirely sure.
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