The probability mass function of a random variable $X$ is given by: $$ P(X = r) = \begin{cases} \frac{{}^nC_r}{32}, & r = 0,1,2,\dots,n \\ 0, & \text{otherwise} \end{cases} $$ Find: $$ P(X \le 2) $$
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Logic Tip: A symmetric binomial distribution (like flipping 5 fair coins) has probabilities that mirror each other: $P(X=0)=P(X=5)$, $P(X=1)=P(X=4)$, etc. Since $P(X \le 2)$ covers exactly half the symmetrical distribution values relative to their weights, it must equal $0.5$.