Question 860804: #13
A survey was conducted at a local ballroom dance studio asking students if they had ever competed in the following dance categories:
A: Smooth
B: Rhythm
C: Standard
The results were then presented to the owner in the following Venn Diagram.
14 is Smooth ONLY (NOT RHYTHM, NOT STANDARD)
31 is Rhythm ONLY (NOT SMOOTH, NOT STANDARD)
30 is Standard ONLY (NOT SMOOTH, NOT RHYTHM)
44 is Smooth and Rhythm ONLY (NOT STANDARD)
18 is Smooth and Standard ONLY (NOT RHYTHM)
39 is Rhythm and Standard ONLY (NOT SMOOTH)
11 is Smooth,Rhythm,Standard ALL THREE
48 is none
If a student is chosen at random, what is the probability that:
Round all answers to 3 decimal places as needed.
A student has competed in none of the categories? is it .256684442?
A student has competed in Smooth, Rhythm, or Standard? is it .4010695187?
A student has competed in Smooth or Standard, but not Rhythm? Is it .0695187166?
A student has competed in Rhythm and Standard, but not Smooth? Is it .2517368984?
A student has competed in Rhythm, given you know they have competed in Smooth. Is it .2406417112?
#14
A survey was conducted at a local ballroom dance studio asking students if they had ever competed in the following dance categories:
A: Smooth
B: Rhythm
C: Standard
The results were then presented to the owner in the following Venn Diagram.
46 is Smooth
36 is Rhythm
11 is Standard
40 is Smooth and Rhythm
22 is Smooth and Standard
16 is Rhythm and Standard
47 is Smooth,Rhythm and Standard
17 is none
If a student is chosen at random, what is the probability that:
Round all answers to 3 decimal places as needed.
A student has competed in none of the categories? Is it .0779816514?
A student has competed in Smooth, Rhythm, or Standard? Is it .4266055046?
A student has competed in Smooth or Standard, but not Rhythm? Is it .0963302752?
A student has competed in Rhythm and Standard, but not Smooth? -.0871559633?
A student has competed in Rhythm, given you know they have competed in Smooth. Is it .376146789?
I COULD NOT PUT IN THE VENN DIAGAM ON HERE FOR BOTH. NEED IT BY APRIL 8TH BY 11:59PM THANK YOU I only get three tries on before problems and these if i get it wrong new problem. ok thank you
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
Notice I have put in the "ONLY"'s below. Otherwise it wouldn't make
sense to have, for instance, 44 as both smooth and rhythm but only 14
as smooth. The 44 that are both smooth and rhythm are
also part of the smooth set. So I will assume you mean the Venn
diagram below. Realize this about the overlapping three circles:
The upper left circle is the SMOOTH circle. It contains 14+44+11+18 = 87
The upper right circle is the RHYTHM circle. It contains 44+31+39+11 = 125
The lower circle is the STANDARD circle. It contains 18+11+39+30 = 98
A survey was conducted at a local ballroom dance studio asking students if they had ever competed in the following dance categories:
A: Smooth
B: Rhythm
C: Standard
The results were then presented to the owner in the following Venn Diagram.
14 is Smooth ONLY (NOT RHYTHM, NOT STANDARD)
31 is Rhythm ONLY (NOT SMOOTH, NOT STANDARD)
30 is Standard ONLY (NOT SMOOTH, NOT RHYTHM)
44 is Smooth and Rhythm ONLY (NOT STANDARD)
18 is Smooth and Standard ONLY (NOT RHYTHM)
39 is Rhythm and Standard ONLY (NOT SMOOTH)
11 is Smooth,Rhythm,Standard (ALL THREE)
48 is none
Here are what the numbers mean:
The 14 are SMMOTH, but NOT rhythm and NOT standard
The 44 are SMOOTH and RHYTHM, but NOT standard
The 31 are RHYTHM, but NOT smooth and not standard
The 18 are SMOOTH and STANDARD, but NOT rhythm
The 11 are SMOOTH, RHYTHM, and STANDARD. (They are inside ALL THREE CIRCLES)
The 39 are RHYTHM and STANDARD, but not SMOOTH
The 30 are STANDARD, but not smooth and not rhythm
The 48 are neither smooth, rhythm, nor standard.
Add all 8 numbers
14+44+31+18+11+39+30+48 = 235 students in the survey.
If a student is chosen at random, what is the probability that:
---------------
A student has competed in none of the categories? is it .256684442?
No it's not. It's the 48 that are outside all three circles -- out
of the 235 total, which is the fraction  = .2042553191
A student has competed in Smooth, Rhythm, or Standard? is it .4010695187?
No it's not. It's all of the 14+44+31+18+11+39+30 = 187 that are in one
of the circles -- out of the 235 total, which is the fraction  =
.7957446809. (You could also get 187 just by subtracting the 48, that are not
in any of the circles, from the 235.)
A student has competed in Smooth or Standard, but not Rhythm? Is it .0695187166?
No it's not. It's the 14+18+30=62 that are in inside either the SMOOTH or
the standard circle but outside the Rhythm circle -- out of the 235 total, which is the fraction  = .2638297872.
A student has competed in Rhythm and Standard, but not Smooth? Is it .2517368984?
No it's not. It's the 31+39+30=100 that are inside either the Rhythm or
the Standard circle but outside the Smooth circle -- out of the 235 total, which is the fraction  = .4255319149.
A student has competed in Rhythm, given you know the student has competed in
Smooth. Is it .2406417112?
No it's not. We are given that the student is one of the 18+11+39+30=98 which
are inside the Smooth Circle. So the probability that the student is also
inside the Rhythm circle, is that he or she is one of 11+44=55 students that
are inside both the Smooth and Rhythm circles.
Answer: 55 out of the 98 or the fraction  = .5612244898.
Now hopefully, you'll study this one and learn how a Venn diagram of three
overlapping circles work, and learn what each of the 8 regions mean, and be
able to do the other one on your own. If you have any questions, ask them
in the thank-you note and I'll get back to you by email, and I'll help you
with the other one. No charge, I just do this for fun.
Edwin
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