On the occasion of the beginning of the winter holidays all of the boys from class 8B went to the shooting range. It is known that there are \(n\) boys in 8B. There are \(n\) targets at the shooting range which the class attended. Each of the boys randomly chooses a target, while some of the boys could choose the same target. After this, all of the boys simultaneously attempt to shoot their target. It is known that each of the boys hits their target. The target is considered to be affected if at least one boy has hit it.
a) Find the average number of affected targets.
b) Can the average number of affected targets be less than \(n/2\)?
\(A\) and \(B\) shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. \(A\) goes first. Find the probability that the shot will occur when \(A\) has the revolver.
Kate and Gina agreed to meet at the underground in the first hour of the afternoon. Kate comes to the meeting place between noon and one o’clock in the afternoon, waits for 10 minutes and then leaves. Gina does the same.
a) What is the probability that they will meet?
b) How will the probability of a meeting change if Gina decides to come earlier than half past twelve, and Kate still decides to come between noon and one o’clock?
c) How will the probability of a meeting change if Gina decides to come at an arbitrary time between 12:00 and 12:50, and Kate still comes between 12:00 and 13:00?
The probability that a purchased lightbulb will work is 0.95. How many light bulbs should I buy so that, with a probability of 0.99, there would be at least 5 that work among them?
A hunter has two dogs. Once, when he was lost in the woods, he went to the fork in the road. The hunter knows that each of the dogs with probability \(p\) will choose the way home. He decided to release the dogs in turn. If both choose the same road, he will follow them; if they are separated, the hunter will choose the road, by throwing a coin. Will this increase the hunter’s chances of choosing the way home, compared to if he had only one dog?
A marketing company decided to carry out a sociological survey to find out which part of the urban population learns news mostly from radio programs, which part – from TV programs, which part – from the press, and which – from the Internet. For the study, it was decided to use a sample of 2,000 randomly chosen owners of email addresses. Can this sample be considered representative?
In a box of 2009 socks there are blue and red socks. Can there be some number of blue socks that the probability of pulling out two socks of the same colour at random is equal to 0.5?
Hannah and Emma have three coins. On different sides of one coin there are scissors and paper, on the sides of another coin – a rock and scissors, on the sides of the third – paper and a rock. Scissors defeat paper, paper defeats rock and rock wins against scissors. First, Hannah chooses a coin, then Emma, then they throw their coins and see who wins (if the same image appears on both, then it’s a draw). They do this many times. Is it possible for Emma to choose a coin so that the probability of her winning is higher than that of Hannah?
Gabby and Joe cut rectangles out of checkered paper. Joe is lazy; He throws a die once and cuts out a square whose side is equal to the number of points that are on the upwards facing side of the die. Gabby throws the die twice and cuts out a rectangle with the length and width equal to the numbers which come out from the die. Who has the mathematical expectation of the rectangle of a greater area?
An exam is made up of three trigonometry problems, two algebra problems and five geometry problems. Martin is able to solves trigonometry problems with probability \(p_1 = 0.2\), geometry problems with probability \(p_2 = 0.4\), and algebra problems with probability \(p_3 = 0.5\). To get a \(B\), Martin needs to solve at least five problems, where the grades are as follows \((A+, A, B, C, D)\).
a) With what probability does Martin solve at least five problems?
Martin decided to work hard on the problems of any one section. Over a week, he can increase the probability of solving the problems of this section by 0.2.
b) What section should Martin complete, so that the probability of solving at least five problems becomes the greatest?
c) Which section should Martin deal with, so that the mathematical expectation of the number of solved problems becomes the greatest?