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 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.
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.
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?
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 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.
The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions (or sectors), numbered from 1 to 13, 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 this sector has already come up previously, and the envelope is no longer there, then the next clockwise sector is played. If it is also empty, then the next one is played, etc., until there is a non-empty sector.
Before the break, the players played six sectors.
a) What is more likely: that sector number 1 has already been played or that sector number 8 has already been plated?
b) Find the probability that, before the break, six sectors with numbers from 1 to 6 were played consecutively.