Question

The p.m.f. of a random variable $X$ is $P(X = x) = \frac{1}{2^5} \binom{5}{x}, x = 0, 1, 2, 3, 4, 5$, then

• A. $P(X \le 2) < P(X \ge 3)$
• B. $P(X \le 2) > P(X \ge 3)$
• C. $P(X \le 2) = 2P(X \ge 3)$
• D. $P(X \le 2) = P(X \ge 3)$

Hint:

For any binomial distribution where the probability of success is exactly $p = 0.5$, the distribution is perfectly symmetric like a mirror. You don't need to calculate any factorials or powers; just pair up the opposite ends!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the probability mass function (p.m.f.) of a binomial distribution and need to compare the cumulative probabilities of the lower half and upper half of the outcomes.

Step 2: Detailed Explanation:
The given p.m.f. represents a binomial distribution with parameters $n = 5$ and $p = \frac{1}{2}$, because it perfectly matches the standard formula $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$:
$$P(X = x) = \binom{5}{x} \left(\frac{1}{2}\right)^x \left(\frac{1}{2}\right)^{5-x} = \frac{1}{2^5} \binom{5}{x}$$ Because $p = \frac{1}{2}$, the distribution is perfectly symmetrical. This means the probability of $x$ successes is equal to the probability of $5-x$ successes:
$$P(X = x) = P(X = 5-x)$$ Let's evaluate the two expressions being compared:
$$P(X \le 2) = P(X = 0) + P(X = 1) + P(X = 2)$$ $$P(X \ge 3) = P(X = 3) + P(X = 4) + P(X = 5)$$ Using the symmetry property:
$P(X = 2) = P(X = 3)$
$P(X = 1) = P(X = 4)$
$P(X = 0) = P(X = 5)$
Since each corresponding term is strictly equal, the total sums must also be equal:
$$P(X \le 2) = P(X \ge 3)$$

Step 3: Final Answer:
The probabilities are strictly equal, matching option (D).