GATE CS 2019 Q.57 (CS Q.47): Probability and Statistics

Suppose $Y$ is distributed uniformly in the open interval $(1, 6)$ . The probability that the polynomial $3x^2 + 6xY + 3Y + 6$ has only real roots is (rounded off to 1 decimal place) ________.

Your answer

Official answer and solution

Official answer: 0.8
From: Official GATE 2019 answer key

Solution

Find the condition on $Y$ for the quadratic equation to have real roots using its discriminant, and evaluate the probability under the uniform distribution on $(1, 6)$ .

The given polynomial is a quadratic equation in $x$ :

3x^2 + 6Yx + (3Y + 6) = 0

Here, the coefficients are $A = 3$ , $B = 6Y$ , and $C = 3Y + 6$ .

A quadratic equation has only real roots if and only if its discriminant $\Delta \ge 0$ :

\Delta = B^2 - 4AC = (6Y)^2 - 4(3)(3Y + 6) = 36Y^2 - 36Y - 72

Setting $\Delta \ge 0$ and dividing by 36:

Y^2 - Y - 2 \ge 0 \implies (Y - 2)(Y + 1) \ge 0

Since $Y$ is distributed in the open interval $(1, 6)$ , the term $(Y + 1)$ is strictly positive ( $Y + 1 > 2 > 0$ ). Thus, the inequality reduces to:

Y - 2 \ge 0 \implies Y \ge 2

The variable $Y \sim \text{Uniform}(1, 6)$ has probability density function $f(y) = \frac{1}{6 - 1} = \frac{1}{5}$ for $y \in (1, 6)$ . The interval where real roots exist is $[2, 6)$ .

The probability is:

P(Y \ge 2) = \frac{6 - 2}{6 - 1} = \frac{4}{5} = 0.8

Final answer

0.8

Common trap

Taking the denominator as 6 instead of $6 - 1 = 5$ , giving $4/6 \approx 0.67$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2019 Computer Science and Information Technology (CS)
Question: 57 of 65, printed as CS Q.47
Section: Computer Science and Information Technology

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