ynaikavde wrote:

A bag contains 15 wool scarves, exactly one of which is red and exactly one of which is green. If Deborah reaches in and draws three scarves, simultaneously and at random, what is the probability that she selects the red scarf but not the green scarf?

a) 2/35

b) 1/15

c) 6/35

d) 13/70

e) 1/5

Theory:

To Find Probability when a red scarf is chosen but a green scarf is not chosen.

First Find probability when Red scarf is necessarily chosen. say this is (1)

Then Find Probability when both of them are chosen together. Say this is (2)

(1) - (2) will give the probability when Red is chosen but Green is not chosen

[ Theory: P(AUB`) = P(A) - P(A intersection B)]Probability of selecting 3 scarves such that a Red scarf is necessarily selected P(R)

= \(\frac{14C2}{15C3}\) (We always choose the red scarf, no restrictions on other two)

= (\(\frac{14!}{12!2!}\)) divided by (\(\frac{15!}{12!3!}\))

= \(\frac{14!12! 3!}{12!2! 15!}\)

= \(\frac{3}{15}\)

= \(\frac{1}{5}\)

Probability of selecting 3 scarves such that One of them is Red and One is a Green scarf

= \(\frac{13C1}{15C3}\) (We choose the red scarf, We choose the green scarf, then out of remaining 13 we choose remaining 1 scarf)

= (\(\frac{13!}{12!}\)) divided by (\(\frac{15!}{12!3!}\))

= \(\frac{13!12! 3!}{15!12!}\)

= \(\frac{13!3!}{15!}\)

= \(\frac{6}{15*14}\)

= \(\frac{1}{35}\)

Now we have to find the Probability that a Red Scarf is selected but a Green scarf is not selected.This will be [ Probability when a Red Scarf is definitely chosen - Probability when a Red Scarf+Green Scarf are chosen together]

= \(\frac{1}{5}\) - \(\frac{1}{35}\)

= \(\frac{7}{35}\) - \(\frac{1}{35}\)

= \(\frac{6}{35}\) C is the answer.