In his laboratory, the Scattered Scientist created a unicellular organism, which, with a probability of 0.6 is divided into two of the same organisms, and with a probability of 0.4 dies without leaving any offspring. Find the probability that after a while the Scattered Scientist will not have any such organisms.
Hercules meets the three-headed snake, the Lernaean Hydra and the battle begins. Every minute, Hercules cuts one of the snake’s heads off. With probability \(\frac 14\) in the place of the chopped off head grows two new ones, with a probability of \(1/3\), only one new head will grow and with a probability of \(5/12\), not a single head will appear. The serpent is considered defeated if he does not have a single head left. Find the probability that sooner or later Hercules will beat the snake.
Anna is waiting for the bus. Which event is most likely?
\(A =\{\)Anna waits for the bus for at least a minute\(\}\),
\(B = \{\)Anna waits for the bus for at least two minutes\(\}\),
\(C = \{\)Anna waits for the bus for at least five minutes\(\}\).
In the set \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), replace one number with two other integers so that the set variance and its mean remain unchanged.
Henry wrote a note on a piece of paper, folded it two times, and wrote “FOR MOM” on the top. Then he unfolded the note, added something to it, randomly folded the note along the old folding lines (not necessarily in the same way as he did it before) and left it on the table with random side up. Find the probability that “FOR MOM” is still on the top.
Alice has six magic pies in her pocket: two magnifying pies (if you eat it, you will grow), and two reducing pies (if you eat it, you will shrink). When Alice met Mary Ann, she, without looking, took out three pies from her pocket and gave them to Mary Ann. Find the probability that one of the girls does not have any magnifying pies.
Prince Charming, and another 49 men and 50 women are randomly seated around a round table. Let’s call a man satisfied, if a woman is sitting next to him. Find:
a) the probability that Prince Charming is satisfied;
b) the mathematical expectation of the number of satisfied men.
Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.
a) What is the fourth number written on the board?
b) What is the eighth number written on the board?
A cube is created from 27 playing blocks.
a) Find the probability that there are exactly 25 sixes on the surface of the cube.
b) Find the probability that there is at least one 1 on the surface of the cube.
c) Find the mathematical expectation of the number of sixes on the surface of the cube.
d) Find the mathematical expectation of the sum of the numbers that are on the surface of the cube.
e) Find the mathematical expectation of a random variable: “The number of different digits that are on the surface of the cube.”
On a calculator, there are numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any key. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last push. The Scattered Scientist pressed a lot of buttons in a random sequence. Find approximately the probability with which the outcome of the resulting chain of actions is an odd number?