James bought \(n\) pairs of identical socks. For \(n\) days James did not have any problems: every morning he took a new pair of socks out of the closet and wore it all day. After \(n\) days, James’ father washed all of the socks in the washing machine and put them into pairs in any way possible as, we repeat, all of the socks are the same. Let’s call a pair of socks successful, if both socks in this pair were worn by James on the same day.
a) Find the probability that all of the resulting pairs are successful.
b) Prove that the expectation of the number of successful pairs is greater than 0.5.
\(N\) pairs of socks hang on a washing line in a random order (the order in which they were taken out of the washing machine). There are no two identical pairs. The socks hang under the drying sheet, so the Scattered Scientist takes out one toe by touch and compares each new sock with all of the previous ones. Find the mathematical expectation of the number of socks taken at the moment when the Scientist will have some pair.
We will assume that the birth of a girl and a boy is equally probable. It is known that in some family there are two children.
a) What is the probability that one of them is a boy and one a girl?
b) Additionally, it is known that one of the children is a boy. What is the probability that there is one boy and one girl in the family now?
c) Additionally, it is known that the boy was born on a Monday. What is the probability that there is one boy and one girl in the family now?
The probability of the birth of twins in Cambria is \(p\), and no triplets are born in Cambria.
a) Evaluate the probability that a random Cambrian that one meets on the street is one of a pair of twins?
b) There are three children in a random Cambrian family. What is the probability that among them there is a pair of twins?
c) In Cambrian schools, twins must be enrolled in the same class. In total, there are \(N\) first-graders in Cambria.
What is the expectation of the number of pairs of twins among them?
There is a deck of playing cards on the table (for example, in a row). On top of each card we put a card from another deck. Some cards may have coincided. Find:
a) the mathematical expectation of the number of cards that coincide;
b) the variance of the number of cards that coincide.
There are 9 street lamps along the road. If one of them does not work but the two next to it are still working, then the road service team is not worried about it. But if two lamps in a row do not work then the road service team immediately changes all non-working lamps. Each lamp does not work independently of the others.
a) Find the probability that the next replacement will include changing 4 lights.
b) Find the mathematical expectation of the number of lamps that will have to be changed on the next replacement.
For the anniversary of the London Mathematical Olympiad, the mint coined three commemorative coins. One coin turned out correctly, the second coin on both sides had two heads, and the third had tails on both sides. The director of the mint, without looking, chose one of these three coins and tossed it at random. She got heads. What is the probability that the second side of this coin also has heads?
A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way (see the figure).
a) \(k\) \(L\)-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.
b) \(7\) \(G\)-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.
Investigating one case, the investigator John Smith discovered that the key witness is the one from the Richardson family who, on that fateful day, came home before the others. The investigation revealed the following facts.
1. The neighbour Maria Ramsden, wanting to borrow some salt from the Richardson’s, rang their doorbell, but no one opened the door. At what time though? Who knows? It was already dark...
2. Jill Richardson came home in the evening and found both children in the kitchen, and her husband was on the sofa – he had a headache.
3. The husband, Anthony Richardson, declared that, when he came home, immediately sat down on the sofa and had a nap. He did not see anyone, nor did he hear anything, and the neighbour definitely did not come – the doorbell would have woken him up.
4. The daughter, Sophie, said that when she returned home, she immediately went to her room, and she does not know anything about her father, however, in the hallway, as always, she stumbled on Dan’s shoes.
5. Dan does not remember when he came home. He also did not see his father, but he did hear how Sophie got angry about his shoes.
“Aha,” thought John Smith. “What is the likelihood that Dan returned home before his father?”.
A toy cube is symmetrical, but it’s unusual: two faces have two points, and the other four have one point. Sarah threw the cube several times, and as a result, the sum of all of the points was 3. Find the probability that one throw resulted in the face with 2 points coming up.