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?
According to the rules of a chess match, the winner is declared to be the one who has beaten their opponent by two defeats. Draws do not count. The probability of winning for both rivals is the same. The number of successful games played in such a match is random. Find its mathematical expectation.
In competitions on stuffing bellies, the chances of opponents winning are the same as the masses of their bodies. Harry weighs more than Will, and Connor weighs less than Sam. It is not possible to draw in such a duel. Harry and Will take turns to compete with Connor and Sam. Which of these events is more likely: \(A = \{\)Harry will win against only Connor, and Will only against Sam\(\}\) or \(B = \{\)Harry will win only against Sam and Will only wins against Connor\(\}\)?
A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.