Q.For a binomial distribution, $n = 5$. If $P(X = 4) = P(X = 3)$ then $p = \ldots$ (A) $\dfrac{1}{3}$ (B) $\dfrac{3}{4}$ (C) $1$ (D) $\dfrac{2}{3}$

When a random variable $X$ counts the number of successes in $n$ independent Bernoulli trials, each with the same success probability $p$ (and failure probability $q=1-p$), $X$ is said to follow a binomial distribution with parameters $n$ and $p$, written $X\sim B(n,p)$. Its probability mass function gives the chance of exactly $x$ successes as $P(X=x)={}^{n}C_x\,p^xq^{n-x}$ for $x=0,1,2,\ldots,n$, where ${}^{n}C_x=\dfrac{n!}{x!(n-x)!}$ counts the number of ways to choose which $x$ of the $n$ trials were the successful ones, and $p^xq^{n-x}$ is the probability of any one such arrangement (by independence). These $(n+1)$ probabilities are exactly the successive terms of the binomial expansion of $(q+p)^n$, which is why they automatically sum to 1. This p.m.f. is the workhorse formula behind every 'find the probability of exactly / at least / at most so many successes' question: 'exactly $x$' uses $P(X=x)$ directly; 'at least' / 'at most' sum several consecutive terms, often more easily via the complement $1-P(\text{the other side})$.