Question

{"given":"One hundred identical coins are tossed, each with probability $p$ of showing heads. Let $X$ be the number of coins showing heads. $X$ follows binomial distribution with parameters $n = 100$ and probability $p$. We are given that $P(X = 50) = P(X = 51)$.","key_observation":"For a binomial distribution, when two consecutive probabilities are equal, we can set up an equation using the binomial probability formula. The key insight is that $P(X = k) = \\binom{n}{k}p^k(1-p)^{n-k}$, and when $P(X = 50) = P(X = 51)$, we can use the relationship between consecutive binomial coefficients to solve for $p$.","option_analysis":[{"label":"(A)","text":"$\\frac{1}{2}$","verdict":"incorrect","explanation":"If $p = \\frac{1}{2}$, the binomial distribution would be symmetric around the mean $np = 50$, but this doesn't satisfy the condition $P(X = 50) = P(X = 51)$ for $n = 100$."},{"label":"(B)","text":"$\\frac{49}{101}$","verdict":"incorrect","explanation":"This value would make the mean $np = \\frac{4900}{101} < 50$, which would not result in equal probabilities at $X = 50$ and $X = 51$ based on the binomial distribution properties."},{"label":"(C)","text":"$\\frac{50}{101}$","verdict":"incorrect","explanation":"This value gives mean $np = \\frac{5000}{101} \\approx 49.5$, which is close to but not exactly the value needed for $P(X = 50) = P(X = 51)$."},{"label":"(D)","text":"$\\frac{51}{101}$","verdict":"correct","explanation":"Setting up the equation $P(X = 50) = P(X = 51)$ and using the relationship between consecutive binomial probabilities, we get $\\frac{1-p}{p} = \\frac{51}{50}$, which gives $p = \\frac{51}{101}$."}],"answer":"(D)","formula_steps":[]}