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
Show 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!
Step 1: Recognise the distribution.
The rule is $P(X=x) = \frac{1}{2^5}\binom{5}{x}$. This is the binomial distribution with $n = 5$ trials and success chance $p = \frac{1}{2}$.
Step 2: See why $p = \frac{1}{2}$ matters.
When $p = \frac{1}{2}$, the distribution is a perfect mirror. The chance of $x$ successes equals the chance of $5 - x$ successes.
\[ P(X=x) = P(X=5-x) \]
Step 3: Write the two sides to compare.
\[ P(X \le 2) = P(0) + P(1) + P(2) \]
\[ P(X \ge 3) = P(3) + P(4) + P(5) \]
Step 4: Pair the terms using the mirror.
$P(2) = P(3)$, $P(1) = P(4)$, and $P(0) = P(5)$.
Step 5: Add the matched pairs.
Since each matching term is equal, the two sums must be equal.
\[ P(X \le 2) = P(X \ge 3) \]
Step 6: Conclusion.
No factorials are needed; the symmetry at $p = 0.5$ does all the work.
\[ \boxed{P(X \le 2) = P(X \ge 3) \text{ (Option 4)}} \]