Two different children are to be selected at random from a group of 12 students. If the probability that both students selected are girls is greater than 
, there must be at least how many girls among the 12 students?
The probability of selecting two girls is
.
There are
ways to select 2 students from the group of 12. Notice that these may be students of either gender; this is the total number of possibilities. If you didn’t know the formula, you might reason this way: there are 12 possibilities for the first student selected, and 11 remaining for the second student selected. Divide by 2 because picking student A and then student B has the same end result as picking student B and then student A, so 12 × 11 double counts each pair.
How many ways are there to select 2 girls from the group? That depends on how many girls there are. If
g is the number of girls, there are
g possibilities for the first girl selected, and
g – 1 possibilities for the second girl selected. And again,
g(
g – 1) double counts each possible pair, so divide by 2.
# of ways to select 2
girls from this group =
.
The probability of selecting 2 girls is greater than
if
, or
g(
g – 1) > 66. Plug the choices to quickly check.
(A) 3(2) = 6
(B) 4(3) = 12
(C) 6(5) = 30
(D) 8(7) = 56
(E) 9(8) = 72
There must be at least 9 girls among the 12 students in order to have greater than
probability of selecting 2 students that are both girls.
The correct answer is E.
