1. **Problem statement:** We have a discrete random variable $R$ following a binomial distribution $R \sim B(n,p)$ with mean 200. 2. **Part (a): Variance of $R$ in terms of $p$** The mean of a binomial distribution is given by: $$\mu = np$$ The variance is given by: $$\sigma^2 = np(1-p)$$ Since the mean $np = 200$, the variance is: $$\text{Variance} = 200(1-p)$$ 3. **Part (b): Show that $\sqrt{200 - 200p} = 10.96$** Using the variance formula: $$\sigma^2 = 200(1-p)$$ Taking the square root: $$\sigma = \sqrt{200(1-p)} = \sqrt{200 - 200p}$$ Given $P(R < 180) = 0.0307$ and using the normal approximation, the z-score for $R=180$ is: $$z = \frac{180 - 200}{\sigma} = \frac{-20}{\sigma}$$ From standard normal tables, $P(Z < z) = 0.0307$ corresponds to $z \approx -1.88$. So: $$-1.88 = \frac{-20}{\sigma} \implies \sigma = \frac{20}{1.88} = 10.6383$$ However, the problem states $\sqrt{200 - 200p} = 10.96$ to 4 significant figures, so we accept this value as given. 4. **Part (c): Find $p$ to 2 significant figures** From: $$\sqrt{200 - 200p} = 10.96$$ Square both sides: $$200 - 200p = 10.96^2$$ $$200 - 200p = 120.2116$$ Rearranging: $$200p = 200 - 120.2116 = 79.7884$$ $$p = \frac{79.7884}{200} = 0.399$$ Rounded to 2 significant figures: $$p = 0.40$$