3-Circle Venn Diag: 600 seniors, 23 in band, 21 in sports,

The principal of a high school composed of 600 seniors wishes to analyze the characteristics of those seniors in the top ten percent of their class. A Venn diagram was used to analyze three events: A = students in the symphonic band, B = students that played a sport, and C = students affiliated with the SGA.

The principal noticed that 23 seniors are in the band, 21 seniors are playing sports, 22 and affiliated with the SGA, 8 are both in the band and playing sports, 5 are both in the band and affiliated with SGA, 9 are playing sports and affiliated with SGA, and 1 student (the valedictorian) is involved in all three activities.

1: If a student from the top ten percent is randomly selected for a qualitative interview, what is the probability that he or she would be affiliated with band given that he or she is affiliated with at least two of the three activities?

i thought this was 14/23, because of P(A intersect B) / P(B) , but that wasn't right

2: If a student from the top ten percent is randomly selected for a qualitative interview, what is the probability that he or she would be affiliated with only one of these three activities?

i thought this was 21 + 22 + 23 - (8 + 1 + 5 + 9) = 43 /60, but thats not right

3: If a student from the top ten percent is randomly selected for a qualitative interview, what is the probability that he or she would be affiliated with none of these activities?

i thought this was 0, because they are all involved according to the venn diagram, but that wasnt right

4: If a student from the top ten percent is randomly selected for a qualitative interview, what is the probability that he or she would be affiliated with band and SGA, but not with sports?

i didnt know how to do this one

5: If a student from the top ten percent is randomly selected for a qualitative interview, what is the probability that he or she would be affiliated with band given that he or she is not affiliated with sports?

clueless on this one too

Re: 3-Circle Venn Diagram problems

Hello, bugmento!

The principal of a high school composed of 60 seniors wishes to analyze their activities.
A Venn diagram was used to analyze three events:
. . A = band
. . B = sports
. . C = SGA

The principal noticed that:
. . 23 are in A
. . 21 are in B
. . 22 are in C
. . 8 are in A and B
. . 5 are in A and C
. . 9 are in B and C
. . 1 is in A, B and C.

Click to expand...

First, construct the Venn diagram . . .

A student is selected at random.
What is the probability that:

1) he is in A, given that he is in at least two of the three activities?

Click to expand...

There are: .\(\displaystyle 7 + 4 + 1 + 8 \:=\:20\) in at least two activities.
Among them, \(\displaystyle 7 + 1 + 4 \:=\:12\) are in A.

\(\displaystyle \text{Therefore: }\;P(\text{in A }|\text{ at least two}) \;=\;\frac{12}{20} \;=\;\frac{3}{5}\)

2) he is in exactly one of the three activities?

Click to expand...

There are: .\(\displaystyle 7 + 4 + 8 \:=\:19\) in exactly one activity.

\(\displaystyle \text{Therefore: }\;P(\text{exactly one}) \;=\;\frac{19}{60}\)

3) he is in none of these activities?

Click to expand...

There are 15 who are in no activity.

\(\displaystyle \text{Therefore: }P(\text{no activity}) \;=\;\frac{15}{60} \:=\:\frac{1}{4}\)

4) he would be in A and C, but not in B?

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There are 4 who are in A and C, but not in B.

\(\displaystyle \text{Therefore: }P(\text{A and C, not B}) \;=\;\frac{4}{60} \;=\;\frac{1}{15}\)

5) he is in A, given that he is not in B?

Click to expand...

\(\displaystyle \text{There are: }\:60 - 21 \:=\:39\) who are not in B.
Among them, \(\displaystyle 11+4 \:=\:15\) are in A.

\(\displaystyle \text{Therefore: }\(\text{A }|\text{ not B}) \;=\;\frac{15}{39} \;=\;\frac{5}{13}\)

Re: 3-Circle Venn Diagram problems

oooooh ok, i see now that i was drawing the Venn Diagram all wrong in the first place.

thanks so much for your help, but just one question about #2, the one where its P(only 1 of the three):

wouldn't this be 11+ 5 + 9 = 25 / 60?

you said it was 7 + 4 + 8 = 19 / 60 -- isn't this one for exactly 2 of the activities?

thanks