In a box, there are 10 white and 15 black balls. Four balls are removed from the box. What is the probability that all of the removed balls will be white?
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?
On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any keys. 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 click.
a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?
b) Solve the previous problem if the multiplication symbol button is repaired.
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.
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.
The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions that are thought up and sent in by the viewers of the programme. Envelopes with the questions are selected in turn in random order with the help of a spinning top with an arrow. If the experts answer correctly, they earn a point, and if they answer incorrectly, the viewers get one point. The game ends as soon as one of the teams scores 6 points. The probability of the team of experts winning in one round is 0.6 and there can be no draws. Currently, the experts are losing 3 to 4. Find the probability that the experts will still win.
King Arthur has two equally wise advisers – Merlin and Percival. Each of them finds the correct answer to any question with probability \(p\) or an incorrect answer, with probability \(q = 1 - p\).
If both counsellors say the same thing, the king listens to them. If they say opposite things, then the king chooses a solution by tossing a coin.
One day, Arthur thought about why he had two advisers, would one not be enough? Then the king called for his counsellors and said:
“It seems to me that the probability of making the right solutions will not decrease if I keep one adviser and listen to him. If so, I must fire one of you. If not, I’ll leave it as it is. Tell me, should I fire one of you?”.
“Who exactly are you going to fire, King Arthur?”, asked the advisers.
“If I make the solution to fire one of you, I will make a choice by tossing a coin”.
The advisers went to think about the answer. The advisors, we repeat, are equally wise, but not equally honest. Percival is very honest and will try to give the right answer, even if he faces dismissal. And Merlin, honest about everything else, in this situation decides to give such an answer with which the probability of his dismissal is as low as possible. What is the probability that Merlin will be fired?
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.
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.