JEE Advance - Mathematics (2020 - Paper 1 Offline - No. 5)
Let C1 and C2 be two biased coins such that the probabilities of getting head in a single toss are $${{2 \over 3}}$$ and $${{1 \over 3}}$$, respectively. Suppose $$\alpha $$ is the number of heads that appear when C1 is tossed twice, independently, and suppose $$\beta $$ is the number of heads that appear when C2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x2 $$-$$ ax + $$\beta $$ are real and equal, is
$${{40} \over {81}}$$
$${{20} \over {81}}$$
$${{1} \over {2}}$$
$${{1} \over {4}}$$
Explanation
It is given that $$\alpha $$ is the number of heads that appear when C1 is tossed twice, the probability distribution of random variable $$\alpha $$ is
Similarly, it is given that $$\beta $$ is the number of heads that appear when C2 is tossed twice, so probability distribution of random variable $$\beta $$ is
Now, as the roots of quadratic polynomial x2 $$-$$ $$\alpha $$x + $$\beta $$ are real and equal, so D = $$\alpha $$2 $$-$$ 4$$\beta $$ = 0 and it is possible if ($$\alpha $$, $$\beta $$) = (0, 0) or (2, 1)
$$ \therefore $$ Required probability
= $${\left( {{1 \over 3}} \right)^2}{\left( {{2 \over 3}} \right)^2} + {\left( {{2 \over 3}} \right)^2}\left( {{4 \over 9}} \right)$$
= $${{4 \over {81}}}$$ + $${{16 \over {81}}}$$ = $${{20 \over {81}}}$$
Similarly, it is given that $$\beta $$ is the number of heads that appear when C2 is tossed twice, so probability distribution of random variable $$\beta $$ is
Now, as the roots of quadratic polynomial x2 $$-$$ $$\alpha $$x + $$\beta $$ are real and equal, so D = $$\alpha $$2 $$-$$ 4$$\beta $$ = 0 and it is possible if ($$\alpha $$, $$\beta $$) = (0, 0) or (2, 1)
$$ \therefore $$ Required probability
= $${\left( {{1 \over 3}} \right)^2}{\left( {{2 \over 3}} \right)^2} + {\left( {{2 \over 3}} \right)^2}\left( {{4 \over 9}} \right)$$
= $${{4 \over {81}}}$$ + $${{16 \over {81}}}$$ = $${{20 \over {81}}}$$
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