Q98 / 150 · MHT CET 2026 (11 April Shift 2) · Mathematics

The probability mass function of a random variable X is given by
$P(X = x) = \dfrac{{}^{5}C_x}{2^5}$, for $x = 0, 1, 2, 3, 4, 5$ and $= 0$, otherwise.
Then which of the following is correct?

  • Correct answer
    A
    $p(x \leq 2) = p(x \geq 3)$
  • B
    $p(x \leq 2) > p(x \geq 3)$
  • C
    $p(x \leq 2) < p(x \geq 3)$
  • D
    $p(x \leq 2) = 2p(x \geq 3)$

Solution

The given probability mass function is $P(X = x) = \dfrac{^{5}C_x}{2^5}$.

We evaluate $P(X \leq 2)$ and $P(X \geq 3)$.

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$

$P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5)$

Using the property of combinations $^{n}C_r = ^{n}C_{n-r}$, we have:

$^{5}C_0 = ^{5}C_5 \Rightarrow P(X = 0) = P(X = 5)$

$^{5}C_1 = ^{5}C_4 \Rightarrow P(X = 1) = P(X = 4)$

$^{5}C_2 = ^{5}C_3 \Rightarrow P(X = 2) = P(X = 3)$

Adding the corresponding terms gives:

$P(X = 0) + P(X = 1) + P(X = 2) = P(X = 3) + P(X = 4) + P(X = 5)$

$P(X \leq 2) = P(X \geq 3)$

Answer: $p(x \leq 2) = p(x \geq 3)$

Hence, the correct answer is option: 'A'.

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