Of the three independent events $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$, the probability that only $E _ { 1 }$ occurs is $\alpha$, only $E _ { 2 }$ occurs is $\beta$ and only $E _ { 3 }$ occurs is $\gamma$. Let the probability $p$ that none of events $E _ { 1 } , E _ { 2 }$ or $E _ { 3 }$ occurs satisfy the equations $( \alpha - 2 \beta ) p = \alpha \beta$ and $( \beta - 3 \gamma ) p = 2 \beta \gamma$. All the given probabilities are assumed to lie in the interval $( 0,1 )$.
$$\text { Then } \frac { \text { Probability of occurrence of } E _ { 1 } } { \text { Probability of occurrence of } E _ { 3 } } =$$

Let $E$, $F$ and $G$ be three events having probabilities $$P(E) = \frac{1}{8}, \quad P(F) = \frac{1}{6}, \quad P(G) = \frac{1}{4},$$ and let $P(E \cap F \cap G) = \frac{1}{10}$.
For any event $H$, if $P(H^c)$ denotes its complement, then which of the following statements is(are) TRUE?
(A) $P(E \cap F \cap G^c) \leq \frac{1}{40}$
(B) $P(E^c \cap F \cap G) \leq \frac{1}{15}$
(C) $P(E \cup F \cup G) \leq \frac{13}{24}$
(D) $P(E^c \cap F^c \cap G^c) \leq \frac{5}{12}$

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac { 1 } { 3 }$, then the probability that the experiment stops with head is
(A) $\frac { 1 } { 3 }$
(B) $\frac { 5 } { 21 }$
(C) $\frac { 4 } { 21 }$
(D) $\frac { 2 } { 7 }$

If $A$ and $B$ are two events such that $P ( A \cup B ) = P ( A \cap B )$, then the incorrect statement amongst the following statements is:
(1) $P ( A ) + P ( B ) = 1$
(2) $P \left( A \cap B ^ { \prime } \right) = 0$
(3) $A \& B$ are equally likely
(4) $P \left( A ^ { \prime } \cap B \right) = 0$

For three events $A$, $B$ and $C$, $P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \dfrac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \dfrac{1}{16}$. Then the probability that at least one of the events occurs, is:
(1) $\dfrac{7}{32}$
(2) $\dfrac{7}{16}$
(3) $\dfrac{1}{64}$
(4) $\dfrac{3}{16}$

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:
(1) $\frac { 127 } { 128 }$
(2) $\frac { 63 } { 64 }$
(3) $\frac { 255 } { 256 }$
(4) $\frac { 1 } { 2 }$

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :
(1) $\frac { 1 } { 6 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 2 } { 3 }$

Four persons can hit a target correctly with probabilities $\frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 }$ and $\frac { 1 } { 8 }$ respectively. If all hit at the target independently, then the probability that the target would be hit, is
(1) $\frac { 25 } { 192 }$
(2) $\frac { 7 } { 32 }$
(3) $\frac { 1 } { 192 }$
(4) $\frac { 25 } { 32 }$

Two newspapers $A$ and $B$ are published in a city. It is known that $25 \%$ of the city population reads $A$ and $20 \%$ reads $B$ while $8 \%$ reads both $A$ and $B$. Further, 30\% of those who read $A$ but not $B$ look into advertisements and $40 \%$ of those who read $B$ but not $A$ also look into advertisements, while $50 \%$ of those who read both $A$ and $B$ look into advertisements. Then the percentage of the population who look into advertisements is:
(1) 13.5
(2) 12.8
(3) 13.9
(4) 13

Let $A , B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $( 1 - k )$, the probability that exactly one of $B$ and $C$ occurs is $( 1 - 2k )$, the probability that exactly one of $C$ and $A$ occurs is $( 1 - k )$ and the probability of all $A , B$ and $C$ occur simultaneously is $k ^ { 2 }$, where $0 < k < 1$. Then the probability that at least one of $A , B$ and $C$ occur is:
(1) greater than $\frac { 1 } { 8 }$ but less than $\frac { 1 } { 4 }$
(2) greater than $\frac { 1 } { 2 }$
(3) greater than $\frac { 1 } { 4 }$ but less than $\frac { 1 } { 2 }$
(4) exactly equal to $\frac { 1 } { 2 }$

Let $E _ { 1 } , E _ { 2 } , E _ { 3 }$ be three mutually exclusive events such that $P \left( E _ { 1 } \right) = \frac { 2 + 3 p } { 6 } , P \left( E _ { 2 } \right) = \frac { 2 - p } { 8 }$ and $P \left( E _ { 3 } \right) = \frac { 1 - p } { 2 }$. If the maximum and minimum values of $p$ are $p _ { 1 }$ and $p _ { 2 }$ then $\left( p _ { 1 } + p _ { 2 } \right)$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 5 } { 4 }$
(4) 1

An amateur archer has 4 arrows and shoots at a balloon placed in the center of a target. The probability of hitting the target on the first shot is 30\%. In successive launches the aim improves, so on the second it is 40\%, on the third 50\% and on the fourth 60\%. It is requested:\ a) (1 point) Calculate the probability that the balloon has burst without needing to make the fourth shot.\ b) (0.5 points) Calculate the probability that the balloon remains intact after the fourth shot.\ c) (1 point) In an exhibition ten professional archers participate, who hit 85\% of their shots. Calculate the probability that among the 10 exactly 6 balloons have burst on the first shot.

Consider two events $A$ and $B$ such that $P ( A ) = 0.5 , P ( B ) = 0.25$ and $P ( A \cap B ) = 0.125$. Answer in a reasoned manner or calculate what is requested in the following cases:\ a) (0.5 points) Let $C$ be another event, incompatible with $A$ and with $B$. Are events $C$ and $A \cup B$ compatible?\ b) (0.5 points) Are $A$ and $B$ independent?\ c) (0.75 points) Calculate the probability $P ( \bar { A } \cap \bar { B } )$ (where $\bar { A }$ denotes the event complementary to event A).\ d) (0.75 points) Calculate $P ( \bar { B } / A )$.

A village mayor election has two polling stations. The proportion of valid votes received by the two candidates at each polling station is shown in the following table (invalid votes are not counted):

	Candidate A	Candidate B
First Polling Station	$40 \%$	$60 \%$
Second Polling Station	$55 \%$	$45 \%$

Assume the number of valid votes at the first and second polling stations are $x$ and $y$ respectively (where $x > 0 , y > 0$), and the candidate with the higher total votes wins. Based on the above table, select the correct options.
(1) When the total number of valid votes $x + y$ is known, the winner can be determined
(2) When the ratio $x : y$ is less than $\frac { 1 } { 2 }$, the winner can be determined
(3) When $x > y$, the winner can be determined
(4) When Candidate A's valid votes at the first polling station exceed those at the second polling station, the winner can be determined
(5) When Candidate B's valid votes at the second polling station exceed those at the first polling station, the winner can be determined

An opaque bag contains blue and green balls, each marked with a number 1 or 2. The quantities are shown in the table below. For example, there are 2 blue balls marked with number 1.

	Blue	Green
Number 1	2	4
Number 2	3	$k$

A ball is randomly drawn from the bag (each ball has an equal probability of being drawn). Given that the event of drawing a blue ball and the event of drawing a ball marked with 1 are independent, what is the value of $k$?
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

There are two parking lots next to a scenic spot. Assume that on a certain day, the probability that either parking lot has no available spaces is 0.7, and whether the two parking lots have available spaces is independent. If a car arrives at these two parking lots on that day, the probability that at least one parking lot has available spaces is 0.(13--1)(13--2).