The random variable $X$ is such that $X ilde{ ext{B}}(n, p)$ The mean value of $X$ is 225 The variance of $X$ is 144 Find $p$ - AQA - A-Level Maths Pure - Question 11 - 2021 - Paper 3

The variance of a binomial distribution is given by:

$ext{Var}(X) = n imes p imes (1 - p)$

Given that the variance is 144, we have:

$144 = n imes p imes (1 - p) ag{3}$

Now, we can substitute Equation (2) into Equation (3).

Letting $p = \frac{225}{n}$:

$144 = n \times \frac{225}{n} \times \left(1 - \frac{225}{n}\right)$

This simplifies to:

$144 = 225 \times \left(1 - \frac{225}{n}\right) = 225 - \frac{225^2}{n}$

Rearranging gives:

$\frac{225^2}{n} = 225 - 144$

$\frac{225^2}{n} = 81$

From here, we can solve for $n$:

$n = \frac{225^2}{81} = 625$

Now substituting $n = 625$ back into Equation (2):

$p = \frac{225}{625} = 0.36$

Thus, the value of $p$ is 0.36. Circle the answer: 0.36.