A die is thrown six times. Find the mathematical expectation of the number of different faces the die lands on.
On a Christmas tree, 100 light bulbs hang in a row. Then the light bulbs begin to switch according to the following algorithm: all are lit up, then after a second, every second light goes out, after another second, every third light bulb changes: if it was on, it goes out and vice versa. After another second, every fourth bulb switches, a second later – every fifth and so on. After 100 seconds the sequence ends. Find the probability that a light bulb straight after a randomly selected light bulb is on (bulbs do not burn out and do not break).
In the final of a contest of performances on March 8 two performances were left. In the first, \(n\) pupils from the class 5A performed, and in the second – \(n\) pupils of class 5B. At the play there were \(2n\) mothers of all \(2n\) students. The best performance is chosen by a vote by the mums. It is known that every mother, with a probability of \(\frac 12\), votes for the best performance and with a probability of \(\frac 12\) – for the performance in which her child participates.
a) Find the probability that the best performance will win with a majority of votes.
b) The same question, if more than two classes have reached the finals.
The mathematics teacher suggested changing the voting scheme at the performance competition. Currently, two groups compete in the final. In the first group, there are \(n\) pupils from class 5A, and in the second, there are \(n\) pupils from class 5B. \(2n\) mothers of all \(2n\) students attended the final of the competition. The best performance is chosen by the mothers voting. It is known that exactly half of the mothers vote honestly for the best performance, and the other half, in any case, vote for the performance in which her child participates (see problem number 65299). The teacher believes that it is necessary to choose a jury of \(2m\) people \((2m \ensuremath{\leq} n)\) from all \(2n\) mums at random. Find the probability that the best performance will win under such voting conditions.
In the Valley of the Five Lakes there are five identical lakes, some of which are connected by streams (in the image, dotted lines denote the possible “routes” of streams). Small carps are born only in lake \(S\). While the carp is growing up, it passes exactly four times from one lake to another by some stream (the carp chooses a stream at random), and then it remains in the lake in which it ended up. Of every thousand carps, an average of 375 remain in lake \(S\), and the rest remain in lake \(B\), no one else lives in the other lakes. Determine how many streams there are in the Valley of the Five Lakes.
A regular dice is thrown many times. Find the mathematical expectation of the number of rolls made before the moment when the sum of all rolled points reaches 2010 (that is, it became no less than 2010).
The bus has \(n\) seats, and all of the tickets are sold to \(n\) passengers. The first to enter the bus is the Scattered Scientist and, without looking at his ticket, takes a random available seat. Following this, the passengers enter one by one. If the new passenger sees that his place is free, he takes his place. If the place is occupied, then the person who gets on the bus takes the first available seat. Find the probability that the passenger who got on the bus last will take his seat according to his ticket?
A fair dice is thrown many times. It is known that at some point the total amount of points became equal to exactly 2010.
Find the mathematical expectation of the number of throws made to this point.
In the city where the Scattered Scientist lives, telephone numbers consist of 7 digits. The scientist easily remembers a phone number, if this number is a palindrome, that is, it is identical when read from left to right and from right to left. For example, the number 4435344 the scientist remembers easily, because this number is a palindrome. And the number 3723627 is not a palindrome, so the scientist does not remember this number easily. Find the probability that the scientist will remember the phone number of a new random acquaintance easily.
A fly crawls along a grid from the origin. The fly moves only along the lines of the integer grid to the right or upwards (monotonic wandering). In each node of the net, the fly randomly selects the direction of further movement: upwards or to the right. Find the probability that at some point:
a) the fly will be at the point \((8, 10)\);
b) the fly will be at the point \((8, 10)\), along the line passing along the segment connecting the points \((5, 6)\) and \((6, 6)\);
c) the fly will be at the point \((8, 10)\), passing inside a circle of radius 3 with center at point \((4, 5)\).