Statistics And Probability Question 489
Question: In a bag there are three tickets numbered 1, 2, 3. A ticket is drawn at random and put back and this is done four times. The probability that the sum of the numbers is even, is
Options:
A) $ \frac{41}{81} $
B) $ \frac{39}{81} $
C) $ \frac{40}{81} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
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The total number of ways of selecting 4 tickets $ =3^{4}=81 $ . The favourable number of ways = sum of coefficients of $ x^{2},\,x^{4},\,....... $ in $ {{(x+x^{2}+x^{3})}^{4}} $ = sum of coefficients of $ x^{2},\,x^{4},\,...... $ in $ x^{4}{{(1+x+x^{2})}^{4}}. $ Let $ {{(1+x+x^{2})}^{4}}=1+a_1x+a_2x^{2}+.....+a_8x^{8}. $ Then $ 3^{4}=1+a_1+a_2+a_3+....+a_8 $ , (On putting $ x=1) $ and $ 1=1-a_1+a_2-a_3+.....+a_8 $ , (On putting $ x=-1) $
$ \therefore ,3^{4}+1=2(1+a_2+a_4+a_6+a_8) $
$ \Rightarrow a_2+a_4+a_6+a_8=41 $ Thus sum of the coefficients of $ x^{2},,x^{4},,……=41 $ Hence the required probaility $ =\frac{41}{81}. $
BETA
