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: Identify the distribution:
\(P(X=x)=\dfrac{\binom5x}{2^5}\) is the binomial pmf with \(n=5\), \(p=\dfrac12\).

Step 2: Compute the two probabilities:
\(P(X\leq2)=\dfrac{1+5+10}{32}=\dfrac{16}{32}\). \(P(X\geq3)=\dfrac{10+5+1}{32}=\dfrac{16}{32}\).

Step 3: Compare:
Both equal \(\dfrac12\). So \(P(X\leq2)=P(X\geq3)\). Option A.

Step 4: Why the other options are wrong.
B and C claim one side is larger, and D claims one side is double. All three contradict the equal values \(\dfrac12\) and \(\dfrac12\).

Final Answer:
p(x <= 2) = p(x >= 3). \[ \boxed{\text{(A) }p(x\leq2)=p(x\geq3)} \]