Question

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

• 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. \(p(x\leq 2) = 2p(x\geq 3)\)

Hint:

Here X is binomial with n = 5 and p = 1/2, so the pmf is symmetric.

Answer 1

The Correct Option is A

Step 1: Use symmetry of binomial coefficients:
$\binom5x=\binom5{5-x}$. So $P(X=x)=P(X=5-x)$.

Step 2: Pair the terms:
$x=0,1,2$ pair with $x=5,4,3$ respectively. So the sum over $\{0,1,2\}$ equals the sum over $\{3,4,5\}$.

Step 3: Pick:
$P(X\leq2)=P(X\geq3)$, option A.

Final Answer:
Binomial coefficients are symmetric, so both sides equal 1/2. \[ \boxed{\text{(A) }p(x\leq2)=p(x\geq3)} \]