Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $29 / 45$, then $n$ is equal to :
(1) 6
(2) 3
(3) 5
(4) 4

There is a lottery consisting of 4 tickets in total: 2 winning tickets and 2 losing tickets. Three people A, B, and C draw one ticket each in that order. Once a ticket is drawn, it is not returned.

60\% of sales in a large department store correspond to items with reduced prices. Customers return 15\% of the discounted items they purchase, a percentage that decreases to 8\% if the items were purchased at full price.
a) (1.25 points) Determine the overall percentage of returned items.
b) ( 1.25 points) What percentage of returned items were purchased at reduced prices?

There are three urns $A$, $B$ and $C$. Urn $A$ contains 4 red balls and 2 black balls, urn $B$ contains 3 balls of each color and urn $C$ contains 6 black balls. An urn is chosen at random and two balls are drawn from it consecutively and without replacement. Find:\ a) (1 point) Calculate the probability that the first ball drawn is red.\ b) (1 point) Calculate the probability that the first ball drawn is red and the second is black.\ c) (0.5 points) Given that the first ball drawn is red, calculate the probability that the second is black.

From a basket with 6 white hats and 3 black hats, one is chosen at random. If the hat is white, a handkerchief is taken at random from a drawer that contains 2 white, 2 black and 5 with white and black checks. If the hat is black, a handkerchief is chosen at random from another drawer that contains 2 white handkerchiefs, 4 black and 4 with white and black checks. It is requested: a) (1 point) Calculate the probability that the handkerchief shows some color that is not the color of the hat. b) (0.5 points) Calculate the probability that in at least one of the accessories (hat or handkerchief) the color black appears. c) (1 point) Calculate the probability that the hat was black, knowing that the handkerchief was checked.

60\% of a sports club's members are women and the remainder are men. This sports club offers the opportunity to play tennis or cricket. Every member plays exactly one of the two sports. $\frac { 2 } { 5 }$ of the male members of the club play cricket; $\frac { 2 } { 3 }$ of the cricketing members of the club are women. What is the probability that a member of the club, chosen at random, is a woman who plays tennis?
A $\frac { 1 } { 5 }$ B $\frac { 7 } { 25 }$ C $\frac { 1 } { 3 }$ D $\frac { 11 } { 25 }$ E $\frac { 3 } { 5 }$

2

Balls are drawn one at a time from a bag containing 3 black balls, 4 red balls, and 5 white balls, and all 12 drawn balls are arranged in a horizontal row in the order they were drawn. Each ball in the bag is equally likely to be drawn.

1. [(1)] Find the probability $p$ that no two red balls are adjacent to each other.
2. [(2)] Given that no two red balls are adjacent to each other, find the conditional probability $q$ that no two black balls are adjacent to each other.

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A fair cubic die is rolled and it is known that one of its faces is in contact with the ground.
Given this, what is the probability that only one of the corners A and B is in contact with the ground?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 5 } { 6 }$

Two different digits are randomly selected from the set $A = \{ 1,2,3,4,5,6,7 \}$.
Given that the product of the selected digits is an even number, what is the probability that the sum of these digits is also an even number?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 5 }$
E) $\frac { 1 } { 6 }$

A certain bus arrives at the bus stop near Duru's house with probability $\dfrac{7}{10}$ at exactly 09:02 and with probability $\dfrac{3}{10}$ at exactly 09:03. Duru leaves home at exactly 09:00 to catch this bus. The time it takes for Duru to reach the stop is 100 seconds with probability $\dfrac{1}{2}$, 150 seconds with probability $\dfrac{3}{10}$, and 250 seconds with probability $\dfrac{1}{5}$.
What is the probability that Duru is at the stop when the bus arrives?
A) $\dfrac{55}{100}$ B) $\dfrac{59}{100}$ C) $\dfrac{63}{100}$ D) $\dfrac{67}{100}$ E) $\dfrac{71}{100}$

In front of the two doors of a shopping mall, there are 2 parking lots named Blue and Red in front of the first door, and 3 parking lots named Yellow, Orange and Green in front of the second door. Kartal, who came to this shopping mall, randomly came in front of one of the doors and randomly parked his car in one of the parking lots in front of that door and entered the shopping mall. When leaving the shopping mall, since Kartal forgot which parking lot he parked his car in and which door he entered the shopping mall from, he exited from one of the doors randomly and searched for his car in one of the parking lots in front of that door randomly.
Accordingly, what is the probability that the parking lot where Kartal searched for his car is the parking lot where he parked his car?
A) $\frac{1}{5}$ B) $\frac{5}{24}$ C) $\frac{6}{25}$ D) $\frac{7}{36}$ E) $\frac{11}{48}$