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

[bugmento]

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

 

[soroban]

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.

First, construct the Venn diagram . . .

      * - - - - - - - - - - - - - - - - - - *
      |   *-----------------*               |
      |   | A               |               |
      |   |           *-----+-----------*   |
      |   |  11       |  7  |         B |   |
      |   |     *-----+-----+-----*     |   |
      |   |     |  4  |  1  |     |  5  |   |
      |   *-----+-----+-----*     |     |   |
      |         |     |        8  |     |   |
      |         |   9 *-----------+-----*   |
      |         | C               |         | 
      |  15     *-----------------*         |
      * - - - - - - - - - - - - - - - - - - *

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?

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?

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?

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?

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?

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

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

 

[bugmento]

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