A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Problem 4 – Numerical Examples

Review statement of Problem 4.

Numerical examples:

$\mathrm{1)}\ \ p = q = \dfrac{1}{2},\ l=1,\ m=2$.

$$\begin{align} (L) &= p(1+q) = 2pq\left(1+ \dfrac{1}{2}\dfrac{p}{q}\right) = \dfrac{3}{4}, \\ (M) &= q^2 = \dfrac{1}{4}. \end{align}$$

$\mathrm{2)}\ \ p=\dfrac{2}{5},\ q= \dfrac{3}{5},\ l=2,\ m=3$.

$$\begin{align} (L) &= p^2\left\lbrace 1+ 2q + 3q^2\right\rbrace = 6p^2q^2\left\lbrace1 + \dfrac{2}{3}\dfrac{p}{q} + \dfrac{1}{6}\dfrac{p^2}{q^2}\right\rbrace = \dfrac{328}{625}. \\ (M) &= q^3 \left\lbrace 1+ 3p \right\rbrace = 4q^3  p\left\lbrace 1 + \dfrac{1}{4}\dfrac{q}{p}\right\rbrace  = \dfrac{297}{625}. \end{align}$$

Continue to Markov's statement of Problem 8.