If an integer $$q$$ be chosen at random in the interval $$ - 10 \leqslant q \leqslant 10,$$ then the probability that the roots of the equation $${x^2} + qx + \frac{{3q}}{4} + 1 = 0$$ are real is :
A.
$$\frac{2}{3}$$
B.
$$\frac{{15}}{{21}}$$
C.
$$\frac{{16}}{{21}}$$
D.
$$\frac{{17}}{{21}}$$
Answer :
$$\frac{{17}}{{21}}$$
Solution :
$$q$$ is an integer, then number of possible outcomes in $$\left[ { - 10,\,10} \right] = 21$$
Now, for real roots, discriminant $$ \geqslant 0$$
$$\eqalign{
& \Rightarrow \left( {q - 4} \right)\left( {q + 1} \right) \geqslant 0 \cr
& \Rightarrow q \geqslant 4,\,\,q \leqslant - 1 \cr} $$
Then, number of favourable outcomes $$= 7 + 10 = 17$$
Hence required probability $$ = \frac{{17}}{{21}}$$
Releted MCQ Question on Statistics and Probability >> Probability
Releted Question 1
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Two events $$A$$ and $$B$$ have probabilities 0.25 and 0.50 respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is 0.14. Then the probability that neither $$A$$ nor $$B$$ occurs is
The probability that an event $$A$$ happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. The probability that the event $$A$$ happens at least once is
If $$A$$ and $$B$$ are two events such that $$P(A) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {\frac{{\overline A }}{{\overline B }}} \right)$$ is equal to
(Here $$\overline A$$ and $$\overline B$$ are complements of $$A$$ and $$B$$ respectively).
A.
$$1 - P\left( {\frac{A}{B}} \right)$$
B.
$$1 - P\left( {\frac{{\overline A }}{B}} \right)$$
C.
$$\frac{{1 - P\left( {A \cup B} \right)}}{{P\left( {\overline B } \right)}}$$
D.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$