Independent Events

Prove that if the events $A_1, \ldots , A_n$ are mutually independent, then they are also pairwise independent.

Prove that if the events $A$ and $B$ are independent, then the following events are also independent:

Prove that if the events $A$ and $B$ are disjoint, with $\mathbb {P}\left(A\right) > 0$ and $\mathbb {P}\left(B\right) > 0$, then the events $A$ and $B$ are dependent.

Suppose the event $A$ is such that it is independent of itself. Prove that then $\mathbb {P}\left(A\right) = 0$ or $\mathbb {P}\left(A\right) = 1$.

Let the events $A$ and $B$ be independent. Prove that if $A \subseteq B$, then $\mathbb {P}\left(A\right) = 0$ or $\mathbb {P}\left(B\right) = 1$.

Determine what the events $A$ and $B$ must be like for the events $AB$ and $A \cup B$ to be independent.

Three coins are tossed in succession. Determine whether the events $A = \left\{ \text{heads appeared on the first coin}\right\}$ and $B = \left\{ \text{at least one tails appeared}\right\}$ are dependent or independent.

A point $\xi = (\xi_1, \xi_2)$ is chosen at random in the square $[0,1]^2$. For which values of $r$ are the events $A = \left\{ \left|\xi_1 - \xi_2\right| \geqslant r\right\}$ and $B = \left\{ \xi_1 + \xi_2 \leqslant 3r\right\}$ independent?

Consider all possible permutations of the four letters $a$, $b$, $c$, and $d$. Determine whether the events $A = \left\{ a \text{ precedes } b\right\}$ and $B = \left\{ c \text{ precedes } d\right\}$ are dependent or independent.

Two dice are rolled. Consider three events: $A$ --- the first die showed an odd number, $B$ --- the second die showed an odd number, $C$ --- the sum of the numbers on the two dice is odd. Check whether the events $A$, $B$, and $C$ are independent:

A point $\xi = (\xi_1, \xi_2)$ is chosen at random in the square $[0,1]^2$. Define three events by $A = \left\{ \xi_1 \leqslant 1/2\right\}$, $B = \left\{ \xi_2 \leqslant 1/2\right\}$, and $C = \left\{ (\xi_1 - 1/2)(\xi_2 - 1/2) \leqslant 0\right\}$. Check whether these events are dependent or independent:

Three dice are rolled. The event $A$ consists in the first and second dice showing the same number of points, $B$ --- the second and third dice showing the same number of points, and $C$ --- the first and third. Check whether the events $A$, $B$, and $C$ are independent:

A tetrahedron is thrown onto a plane; three of its faces are colored red, blue, and green respectively, while the fourth face carries all three colors. The event $K$ means that the face landing on the plane contains the color red, the event $S$ --- that it contains the color blue, and $Z$ --- the color green. Check whether the events $K$, $S$, and $Z$ are independent:

Two fair coins are tossed. Let the event $A$ consist in heads appearing on the first coin, the event $B$ in heads appearing on the second coin, and the event $C$ in heads appearing on exactly one (and only one) of the coins. Check whether the events $A$, $B$, and $C$ are independent:

Let the events $A_1, \ldots , A_n$ be mutually independent with probabilities $\mathbb {P}\left(A_k\right) = p_k$. Find the probabilities:

An electrical circuit consists of elements $A_k$ connected according to the following scheme:

Let $\Omega = [0, 1]$, let $\mathscr {F}$ be the $\sigma$-algebra of Borel sets, let $\mathbb {P}$ be Lebesgue measure, and let $A = [0, 1/2]$. Construct events $B$ and $C$ such that $\mathbb {P}\left(B\right) = \mathbb {P}\left(C\right) = 1/2$, and the events $A$, $B$, and $C$ are mutually independent.

There are 3 pairwise independent events which nevertheless cannot all occur together. Assuming they all have the same probability $p$, determine the largest possible value of $p$.

Let the events $A$, $B$, and $C$ be pairwise independent, with $\mathbb {P}\left(C\right) > 0$. Is the equality $\mathbb {P}\left(A \cup B \mid C\right) = \mathbb {P}\left(A \cup B\right)$ true?

What is the minimum number of points a finite probability space must have in order for $n$ mutually independent events $A_1, \ldots , A_n$, each with probability different from zero and one, to be definable on that space?

The probability that a molecule, having collided with another molecule at time $t_0 = 0$ and having had no further collisions up to time $t_1$, will collide again in the time interval between $t_1$ and $t_1 + \Delta t$ is equal to $\lambda \Delta t + o(\Delta t)$ as $\Delta t \to 0$. Find the probability that the free path time (the time between two successive collisions) will exceed $t$.

Find the probability that a machine, working at time $t_0$, will not stop before time $t_1$, given that:

1. this probability depends only on the length of the time interval $(t_0, t_1)$;

2. a machine working at time $t$ will stop during the time interval $(t, t + \Delta )$ with probability $\alpha \Delta + o(\Delta )$ as $\Delta \to 0$, where $\alpha > 0$.

In constructing mortality tables, one often starts from the following assumptions:

1. a person who has survived to age $t$ will die within the time from $t$ to $t + \Delta$ with probability $p(t, \Delta ) = \alpha (t)\Delta + o(\Delta )$ as $\Delta \to 0$, where $\alpha (t) > 0$;

2. the probability of death at the moment of birth is zero. Under these assumptions, find the probability of death before age $t$.