Conditional Probability

Three dice are thrown. Find the probability that a "six" comes up on all the dice, given that

1. at least one die shows a "six";

2. at least two dice show an equal number of points.

One box contains 1 white ball and 2 black balls, while another box contains 2 white balls and 3 black balls. Two balls chosen at random from the first box and two balls chosen at random from the second box are placed into a third box. Find the probability that

1. a ball chosen at random from the third box will be white;

2. when two balls are drawn without replacement from the third box, one of them will be white and the other black.

From an urn containing $a$ white and $b$ black balls, two players draw balls in turn. The one who draws a white ball first wins. Find the probability that the first player wins, in the cases when the balls are drawn

1. by the scheme of equiprobable sampling with replacement;

2. by the scheme of equiprobable sampling without replacement.

Give examples showing that, in general, the equalities

are false.

A group of 15 people is taking an exam in probability theory. The syllabus contains 31 exam tickets, five of which the students consider "easy". In what order should one enter the room to have the greatest probability of drawing an easy ticket?

During tests of a device at a pasta factory, it was established that the probability of its exploding in the absence of disturbances is 0.01, under overheating -0.05, under vibration -0.1, under vibration and overheating together -0.2. Find the probability of an explosion at the pasta factory when operating in hot countries (the probability of overheating is 0.2, the probability of vibration -0.1, the probability of overheating and vibration together -0.02).

From the collection of all subsets of the set $S=\left\{ 1,2, \ldots , N\right\}$, sets $A_{1}$ and $A_{2}$ are chosen by the scheme of sampling with replacement. Find the conditional probability that the sets $A_{1}$ and $A_{2}$ consist of $l_{1}$ and $l_{2}$ elements respectively, given that they do not intersect.