JEE MAIN - Mathematics (2020 - 9th January Morning Slot - No. 3)
In a box, there are 20 cards, out of which 10
are lebelled as A and the remaining 10 are
labelled as B. Cards are drawn at random, one
after the other and with replacement, till a
second A-card is obtained. The probability that
the second A-card appears before the third
B-card is :
$${{13} \over {16}}$$
$${{11} \over {16}}$$
$${{15} \over {16}}$$
$${{9} \over {16}}$$
Explanation
Possibilities that the second A card appears before the third B card are
=AA + ABA + BAA + ABBA + BBAA + BABA
= $${\left( {{1 \over 2}} \right)^2}$$ + $${\left( {{1 \over 2}} \right)^3}$$ + $${\left( {{1 \over 2}} \right)^3}$$ + $${\left( {{1 \over 2}} \right)^4}$$ + $${\left( {{1 \over 2}} \right)^4}$$ + $${\left( {{1 \over 2}} \right)^4}$$
= $${1 \over 4}$$ + $${1 \over 8}$$ + $${1 \over 8}$$ + $${1 \over {16}}$$ + $${1 \over {16}}$$ + $${1 \over {16}}$$
= $${{11} \over {16}}$$
=AA + ABA + BAA + ABBA + BBAA + BABA
= $${\left( {{1 \over 2}} \right)^2}$$ + $${\left( {{1 \over 2}} \right)^3}$$ + $${\left( {{1 \over 2}} \right)^3}$$ + $${\left( {{1 \over 2}} \right)^4}$$ + $${\left( {{1 \over 2}} \right)^4}$$ + $${\left( {{1 \over 2}} \right)^4}$$
= $${1 \over 4}$$ + $${1 \over 8}$$ + $${1 \over 8}$$ + $${1 \over {16}}$$ + $${1 \over {16}}$$ + $${1 \over {16}}$$
= $${{11} \over {16}}$$
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