The Concept of Independence. The Bernoulli Scheme

Balls are drawn from a box containing black and white balls. Let the event $A_{k}$ mean that a white ball is drawn on the $k$-th step. Prove that the events $A_{1}, \ldots , A_{n}$

1. are mutually independent, if the balls are drawn with replacement;

2. are dependent, if the balls are drawn without replacement.

Two players carry out a series of independent trials. In each trial, player $A$ throws 3 dice, while player $B$ throws 2 dice simultaneously with player $A$ and independently of him. They carry out these trials in succession until the first occurrence of a "six" on at least one of the dice. Find the probabilities of the following events:

1. $\mathcal{C}=\left\{ \text{ the "six" first appeared for player }B\text{, and not for }A\right\}$

2. $\mathcal{D}=\left\{ \text{ the "six" first appeared simultaneously for }A\text{ and }B\right\}$.

Let $A, B, C$ be pairwise independent equiprobable events such that $A \cap B \cap C=\varnothing$. Find the maximum possible value of $\mathbb {P}\left(A\right)$.

A set $S$ of $n$ elements is given. Three subsets $A, B, C$ are chosen from it randomly and independently. Each random subset is formed as follows: each element of the set $S$, independently of the others, is included in the subset with probability $p$, and is not included with probability $(1-p)$ (this process is repeated three times: first for the set $A$, then for $B$, and finally for $C)$. Find the probability of the event $D=\left\{ A \cap B \subseteq C \subseteq A \cup B\right\}$.

A social network user views the wall of one of his 7 friends each day with probability $p_{i}, i=1, \ldots , 7$; $\nu$ is the number of the first week during which he views the walls of all seven of his friends. Find the probability that on the Monday of week $\nu$ he views the wall of his first friend.

The edges of the complete graph $K_{n}$ on $n$ vertices are colored independently of each other, with equal probability $\frac{1}{k}$, in one of $k$ colors. Let $V$ be the set of vertices of the graph $K_{n}$, and $S \subset V$. Denote by $A_{S}$ the following event: $A_{S}=\left\{ \text{ all edges of }K_{n}\text{ both of whose vertices belong to }S\text{ are colored the same color }\right\}$. Under what conditions on the mutual arrangement of the subsets $S, T \subset V$ are the events $A_{S}$ and $A_{T}$ independent?

Let $A_{1}, \ldots , A_{n}$ be independent events. Prove that