An urn contains 30 balls out of which one is special. If 6 of these balls are taken out at random, what is the probability that the special ball is chosen?
(A) $\frac { 1 } { 30 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 5 }$
(D) $\frac { 1 } { 15 }$

2. In a piggy bank there are 15 coins, of which 9 are 1 euro coins and the other 6 are 2 euro coins. 6 coins are drawn simultaneously. – What is the probability that the total value of the coins drawn is exactly 10 euros? – What is the probability that the total value of the coins drawn is at most 10 euros?

19. A bag contains 12 red balls and 6 white balls. Six balls are drawn one by one without replacement of which atleast 4 balls are white. Find the probability that in the next two draws exactly one white ball is drawn. (leave the answer in terms of ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } }$ ).
Sol. Let $\mathrm { P } ( \mathrm { A } )$ be the probability that atleast 4 white balls have been drawn. $P \left( A _ { 1 } \right)$ be the probability that exactly 4 white balls have been drawn. $\mathrm { P } \left( \mathrm { A } _ { 2 } \right)$ be the probability that exactly 5 white balls have been drawn. $P \left( A _ { 3 } \right)$ be the probability that exactly 6 white balls have been drawn. $\mathrm { P } ( \mathrm { B } )$ be the probability that exactly 1 white ball is drawn from two draws. $\mathrm { P } ( \mathrm { B } / \mathrm { A } ) = \frac { \sum _ { \mathrm { i } = 1 } ^ { 3 } \mathrm { P } \left( \mathrm { A } _ { \mathrm { i } } \right) \mathrm { P } \left( \mathrm { B } / \mathrm { A } _ { \mathrm { i } } \right) } { \sum _ { \mathrm { i } = 1 } ^ { 3 } \mathrm { P } \left( \mathrm { A } _ { \mathrm { i } } \right) } = \frac { \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } \cdot \frac { { } ^ { 10 } \mathrm { C } _ { 1 } { } ^ { 2 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } \cdot \frac { { } ^ { 11 } \mathrm { C } _ { 1 } { } ^ { 1 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } } } { \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } + \frac { { } ^ { 12 } \mathrm { C } _ { 0 } { } ^ { 6 } \mathrm { C } _ { 6 } } { { } ^ { 18 } \mathrm { C } _ { 6 } } }$ $= \frac { { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } { } ^ { 10 } \mathrm { C } _ { 1 } { } ^ { 2 } \mathrm { C } _ { 1 } + { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } { } ^ { 11 } \mathrm { C } _ { 1 } { } ^ { 1 } \mathrm { C } _ { 1 } } { { } ^ { 12 } \mathrm { C } _ { 2 } \left( { } ^ { 12 } \mathrm { C } _ { 2 } { } ^ { 6 } \mathrm { C } _ { 4 } + { } ^ { 12 } \mathrm { C } _ { 1 } { } ^ { 6 } \mathrm { C } _ { 5 } + { } ^ { 12 } \mathrm { C } _ { 0 } { } ^ { 6 } \mathrm { C } _ { 6 } \right) }$

There are two bags, A and B. Bag A contains four white balls and one red ball, and bag B contains two white balls and three red balls. Two balls are taken simultaneously out of bag A, then two balls are taken simultaneously out of bag B.
(1) The probability that two white balls are taken out of A, and one white ball and one red ball are taken out of B is $\frac{\mathbf{J}}{\mathbf{J}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{M}}{\mathbf{M}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ PQ
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{RS}}{\mathbf{TU}}$.

There are two bags, A and B. Bag A contains four white balls and one red ball, and bag B contains two white balls and three red balls. Two balls are taken simultaneously out of bag A, then two balls are taken simultaneously out of bag B.
(1) The probability that two white balls are taken out of A, and one white ball and one red ball are taken out of B is $\frac{\mathbf{O}}{\mathbf{PQ}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{R}}{\mathbf{S}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ T UV
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{WX}}{\mathbf{W}}$.

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

2. In the checkerboard grid shown on the right, two squares are randomly selected. The probability that the two selected squares are not in the same row (whether or not they are in the same column is irrelevant) is
(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 3 } { 5 }$
(5) $\frac { 4 } { 5 }$

A bag contains blue, green, and yellow balls totaling 10 balls. Two balls are randomly drawn from the bag (each ball has an equal probability of being drawn). The probability that both balls drawn are blue is $\frac{1}{15}$, and the probability that both are green is $\frac{2}{9}$. The probability that two randomly drawn balls are of different colors is $\frac{\text{(16--1)}}{\text{(16--3)}}$. (Express as a fraction in lowest terms)

Let $n$ be an odd number greater than or equal to 5. Consider a circle centered at point O in the plane, and a regular $n$-gon inscribed in it. Choose 4 distinct points simultaneously from the $n$ vertices. Assume that any 4 points are equally likely to be chosen. Find the probability $p_n$ that the quadrilateral with the chosen 4 points as vertices contains O in its interior.

A bag contains 2 red, 2 white, and 1 yellow marble.
When 4 marbles are randomly drawn from the bag, what is the probability that the remaining marble in the bag is red?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 2 } { 5 }$
E) $\frac { 3 } { 5 }$

From a group of 6 girls and 7 boys, 2 representatives are selected.
What is the probability that one of the two selected representatives is a girl and the other is a boy?
A) $\frac { 3 } { 4 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 2 } { 13 }$
D) $\frac { 7 } { 13 }$
E) $\frac { 9 } { 13 }$

A bag contains 5 red and 4 white marbles.
When 3 marbles are drawn randomly from this bag at the same time, what is the probability that there are at most 2 marbles of each color?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 7 } { 8 }$
E) $\frac { 8 } { 9 }$

Four identical matches are taken, each with only one flammable end. These matches are randomly arranged along all sides of a square whose side length is the same as the length of one match, with the ends touching each other.
What is the probability that there are no flammable ends in contact with each other in this arrangement?
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 1 } { 16 }$
E) $\frac { 3 } { 16 }$

An exam consisting of a total of 8 questions, with 4 questions each in the verbal and quantitative sections, has the following statement in its booklet: ``To pass the exam, you must answer at least 5 questions correctly in total, with at least 2 questions from each of the verbal and quantitative sections.'' Sevcan, who read this statement incompletely, randomly selected 5 out of 8 questions on the exam and answered each question she selected correctly.
Accordingly, what is the probability that Sevcan passes the exam?
A) $\frac{3}{4}$
B) $\frac{4}{5}$
C) $\frac{5}{6}$
D) $\frac{6}{7}$
E) $\frac{7}{8}$

Kerem randomly selects 3 numbers using the buttons shown in the figure to create the password for his locker, such that each is in a different row and different column.
Accordingly, what is the probability that all of the numbers Kerem selected are odd?
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) (from figure)
D) $\frac{5}{9}$
E) $\frac{4}{27}$