Attorneys are questioning 11 potential jurors about their ability to serve on a jury and decide a case fairly. Seven individuals will be selected to serve on the jury. If each potential juror has an equal chance of being selected, what is the probability that Tamara and Inga are chosen for the jury?

A. 2/11
B. 2/7
C. 21/55
D. 4/11
E. 34/55

A direct probabilistic approach:

First person to be chosen one of the two girls - 2/11
Second person to be chosen to be the other girl - 1/10
The probability to chose both girls - (2/11)*(1/10) = 1/55.
The two girls can be chosen anywhere in the list of the 7 jurors, therefore the final probability is (1/55)*7C2 = (1/55)*(7*6/2) = 21/55.

Combinatorial approach:
Total number of possibilities to choose the 7 jurors - 11C7 = 11*10*9*8/(2*3*4) = 11*10*3.
Number of possibilities to have the two girls among the final 7 jurors - 9C5 = 9*8*7*6/(2*3*4) = 9*2*7 (the two girls must be in, therefore we need to choose 5 more people from the remaining 9).
Required probability 9*2*7/(11*10*3) = 21/55.
(I prefer not to multiply before I express the final probability as a ratio, I always try to reduce a fraction whenever possible.)

Answer C.
_________________

Thank you, you are right: it should be 1/55*(7C2). The two girls can be chosen anywhere in the list of the final 7 jurors.
Corrected my original post.
_________________

Hi,

Though i got the answer right, i initially approach did not yield the right answer.
Please help me identify the error in my thought process

Chance that the two girls are selected = 1- (Chance that none of the 2 girls are selected)
i.e - Ways that none of the two girls are selected = 9C7
- Total ways of selecting 7 of 11 people = 11C7

Answer = 1-(9C7/11C7) = 1-(6/55) = 49/55

I chose the combinatorics:
we need to choose 7 people:
7C11 = 55
from 7, we need to have both of them. so 2C7 = 22
probability to be both are thus 22/55

P(Tamara and Inga are BOTH chosen for the jury) = (# of ways to choose 7 people that include Tamara and Inga)/(TOTAL # of ways to choose 7 people)

# of ways to choose 7 people that include Tamara and Inga
Stage 1: place Tamara and Inga on the jury. This can be accomplished in 1 way
Stage 2: select 5 more people from the remaining 9 people. This can be accomplished in 9C5 ways (= 126 ways)
By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create a jury with Tamara and Inga) in (1)(126) ways (= 126 ways)


TOTAL # of ways to choose 7 people
We can choose 7 people from 11 people in 11C7 ways
11C7 = 330

So, P(Tamara and Inga are BOTH chosen for the jury) = 126/330
= 63/165
= 21/55

Answer: C

Cheers,
Brent
_________________

Exactly KanishkM !

I explain to my students that they should think Tamara and Inga "took" two places in advance, so that we have in fact 7-2 = 5 places available, to be occupied from people chosen among 11-2 individuals.

Regards,
Fabio.
_________________