When a missile is fired from a ship, the probability that it is intercepted is $\frac { 1 } { 3 }$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac { 3 } { 4 }$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:
(1) $\frac { 3 } { 8 }$
(2) $\frac { 1 } { 27 }$
(3) $\frac { 1 } { 8 }$
(4) $\frac { 3 } { 4 }$

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :
(1) $\frac { 3 } { 4 }$
(2) $\frac { 52 } { 867 }$
(3) $\frac { 39 } { 50 }$
(4) $\frac { 22 } { 425 }$

The probability distribution of random variable $X$ is given by:

$X$	1	2	3	4	5
$P ( X )$	$K$	$2 K$	$2 K$	$3 K$	$K$

Let $p = P ( 1 < X < 4 \mid X < 3 )$. If $5 p = \lambda K$, then $\lambda$ is equal to

A random variable $X$ has the following probability distribution:

$X$	0	1	2	3	4
$P ( X )$	$k$	$2 k$	$4 k$	$6 k$	$8 k$

The value of $P \left( \frac { 1 < x < 4 } { x \leq 2 } \right)$ is equal to
(1) $\frac { 4 } { 7 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 4 } { 5 }$

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is
(1) $\frac{5}{3}$
(2) $\frac{2}{7}$
(3) $\frac{3}{7}$
(4) $\frac{5}{6}$

$25\%$ of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac { k } { 10 }$. Then the value of $k$ is $\_\_\_\_$.

In a bolt factory, machines $A , B$ and $C$ manufacture respectively $20 \% , 30 \%$ and $50 \%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to be defective then the probability that it is manufactured by the machine $C$ is
(1) $\frac { 5 } { 14 }$
(2) $\frac { 9 } { 28 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 2 } { 7 }$

A company has two plants $A$ and $B$ to manufacture motorcycles. $60\%$ motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. $80\%$ of the motorcycles manufactured at plant $A$ are rated of the standard quality, while $90\%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. Find the probability that it was manufactured at plant $B$.

Bag $A$ contains 3 white, 7 red balls and bag $B$ contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $A$, if the ball drawn is white, is:
(1) $\frac{1}{4}$
(2) $\frac{1}{9}$
(3) $\frac{1}{3}$
(4) $\frac{3}{10}$

There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is : (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 5 } { 12 }$ (4) $\frac { 1 } { 3 }$

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $m + n$ is equal to:
(1) 4
(2) 14
(3) 13
(4) 11

If $A$ and $B$ are two events such that $P ( A \cap B ) = 0.1$, and $P ( A \mid B )$ and $P ( B \mid A )$ are the roots of the equation $12 x ^ { 2 } - 7 x + 1 = 0$, then the value of $\frac { \mathrm { P } ( \overline { \mathrm { A } } \cup \overline { \mathrm { B } } ) } { \mathrm { P } ( \overline { \mathrm { A } } \cap \overline { \mathrm { B } } ) }$ is :
(1) $\frac { 4 } { 3 }$
(2) $\frac { 7 } { 4 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 9 } { 4 }$

Bag $B _ { 1 }$ contains 6 white and 4 blue balls, Bag $B _ { 2 }$ contains 4 white and 6 blue balls, and Bag $B _ { 3 }$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B _ { 2 }$, is :
(1) $\frac { 4 } { 15 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 2 } { 3 }$

Q80. Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :
(1) $\frac { 5 } { 18 }$
(2) $\frac { 5 } { 16 }$
(3) $\frac { 4 } { 17 }$
(4) $\frac { 7 } { 18 }$

Q80. A company has two plants $A$ and $B$ to manufacture motorcycles. $60 \%$ motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B .80 \%$ of the motorcycles manufactured at plant $A$ are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126 p$ is
(1) 54
(2) 66
(3) 64
(4) 56
Q81.Let $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$ be the solution of the equation $4 x ^ { 4 } + 8 x ^ { 3 } - 17 x ^ { 2 } - 12 x + 9 = 0$ and $\left( 4 + x _ { 1 } ^ { 2 } \right) \left( 4 + x _ { 2 } ^ { 2 } \right) \left( 4 + x _ { 3 } ^ { 2 } \right) \left( 4 + x _ { 4 } ^ { 2 } \right) = \frac { 125 } { 16 } m$. Then the value of $m$ is

Q80. There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is :
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 5 } { 12 }$
(4) $\frac { 1 } { 3 }$

According to data from the Diabetes Foundation, 13.8\% of Spanish people over 18 years old have diabetes, although 43\% of them do not know they have it. A Spanish person over 18 years old is chosen at random.
a) (1 point) What is the probability that they are diabetic and know it? What is the probability that they are not diabetic or do not know they are?
b) (1.5 points) A certain test correctly diagnoses 96\% of positive diabetes cases, but gives 2\% false positives. If a Spanish person over 18 years old tests positive, what is the probability that they are actually diabetic?

In an urn there are two white balls and four black balls. A ball is drawn at random. If the ball drawn is white, it is returned to the urn and another white ball is added; if it is black, it is not returned to the urn. Next, a ball is drawn at random from the urn again.
a) (1 point) What is the probability that the two balls drawn are of different colors?
b) (1.5 points) What is the probability that the first ball drawn was black, given that the second was white?

An air quality measurement station measures levels of $\mathrm { NO } _ { 2 }$ and suspended particles. The probability that on a given day a level of $\mathrm { NO } _ { 2 }$ exceeding the permitted level is measured is 0.16. On days when the permitted level of $\mathrm { NO } _ { 2 }$ is exceeded, the probability that the permitted level of particles is exceeded is 0.33. On days when the level of $\mathrm { NO } _ { 2 }$ is not exceeded, the probability that the level of particles is exceeded is 0.08.\ a) ( 0.5 points) What is the probability that on a given day both permitted levels are exceeded?\ b) ( 0.75 points) What is the probability that at least one of the two is exceeded?\ c) ( 0.5 points) Are the events ``on a given day the permitted level of $\mathrm { NO } _ { 2 }$ is exceeded'' and ``on a given day the permitted level of particles is exceeded'' independent?\ d) ( 0.75 points) What is the probability that on a given day the permitted level of $\mathrm { NO } _ { 2 }$ is exceeded, given that the permitted level of particles has not been exceeded?

A company markets three types of products A, B and C. Four out of every seven products are of type A, two out of every seven products are of type B and the rest are of type C. For export, 40\% of type A products, 60\% of type B products and 20\% of type C products are destined. A product is chosen at random, it is requested: a) (1.25 points) Calculate the probability that the product is destined for export. b) (1.25 points) Calculate the probability that it is of type C given that the product is destined for export.

Knowing that $P ( A ) = 0.5 , P ( A / B ) = 0.625$ and $P ( A \cup B ) = 0.65$, find:\ a) ( 1.5 points) $P ( B )$ and $P ( A \cap B )$.\ b) (1 point) $P ( A / A \cup B )$ and $P ( A \cap B / A \cup B )$

According to a certain country's investigation of missing light aircraft: 70\% of missing light aircraft are eventually found. Among the aircraft that are found, 60\% have emergency locator transmitters installed; among the missing aircraft that are not found, 90\% do not have emergency locator transmitters installed. Emergency locator transmitters send signals when the aircraft crashes, allowing rescue personnel to locate it. A light aircraft is now missing. If it is known that the aircraft has an emergency locator transmitter installed, the probability that it will be found is (15--1)(15--2).

A bag contains 100 balls numbered $1, 2, \ldots, 100$ respectively. A person randomly draws one ball from the bag, and each ball has an equal probability of being drawn. Under which of the following conditions is the conditional probability that the person draws ball number 7 the largest?
(1) The number of the ball drawn is odd (2) The number of the ball drawn is prime (3) The number of the ball drawn is a multiple of 7 (4) The number of the ball drawn is not a multiple of 5 (5) The number of the ball drawn is less than 10

All senior high school students at a certain school have taken either Mathematics A or Mathematics B on the scholastic aptitude test. Among these students, those taking only Mathematics A account for $\frac { 3 } { 10 }$ of all senior high school students. Among students taking Mathematics A, $\frac { 5 } { 8 }$ also took Mathematics B. What is the proportion of students taking only Mathematics B among all students at the school taking Mathematics B? (Express as a fraction in lowest terms)

9. Consider the statement about Fred: (}) Every day next week, Fred will do at least one maths problem. If statement (*) is not true, which of the following is certainly true?
A Every day next week, Fred will do more than one maths problem.
B Some day next week, Fred will do more than one maths problem.
C On no day next week will Fred do more than one maths problem.
D Every day next week, Fred will do no maths problems.
E Some day next week, Fred will do no maths problems. F On no day next week will Fred do no maths problems.