Odds in Favor and Odds against an Event

Q. Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g\left(3\right)=2g\left(1\right)$ is

1. $\frac{1}{30}$
2. $\frac{1}{10}$
3. $\frac{1}{15}$
4. $\frac{1}{5}$

Q.

A die is tossed $5$ times. Getting an odd number is considered a success. Then the variance of the distribution of success is

1. $\frac{8}{3}$

2. $\frac{3}{8}$

3. $\frac{4}{5}$

4. $\frac{5}{4}$

Q.

A letter is known to have come either from 'TATA NAGAR' or from 'CALCUTTA'. On the envelope, just two consecutive letters TA are visible. what is the probability that the letter came from 'TATA NAGAR' ?

Q. The probability distribution of random variable $X$ is given by:
$\begin{array}{cccccc}X& 1& 2& 3& 4& 5\\ P\left(X\right)& K& 2K& 2K& 3K& K\end{array}$
Let $p=P(1<X<4|X<3)$. If $5p=\lambda K$, then $\lambda$ is equal to

Q. The odds in favour of A solving a problem are 3 to 4 and the odds against B solving the same problem are 5 to 7. If they both try the problem, the probability that the problem is solved is:

1. $\frac{41}{84}$
2. $\frac{16}{21}$
3. $\frac{5}{21}$
4. $\frac{1}{4}$

Q. Ten cards numbered from $1$ to $10$ are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than $3$, then the probability that it is an even number is :

1. $\frac{4}{9}$
2. $\frac{4}{5}$
3. $\frac{4}{7}$
4. $\frac{5}{7}$

Q.

In a race, the odds in favour of horses A, B, C, D are 1 : 3, 1 : 4, 1 : 5 and

1 : 6 respectively. Find probability that one of them wins the race.

Q.

There are three events A, B, C one of which must and only one can happen. The odds are 8 to 3 against A. 5 to 2 against B, finds the odds against C.

Q. A ship is fitted with three engines $E_1,E_2,\text{and}{E}_3$. The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}\text{and}\frac{1}{4}.$ For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let $X_1,X_2\text{and}{X}_3$ denote respectively the events that the engines $E_1,E_2\text{and}{E}_3$ are functioning. Which of the following is (are) true?

1. $P\left[X_1^c|X\right]=\frac{3}{16}$
2. P[Exactly two engines of the ship are functioning |X] $=\frac{7}{8}$
3. $P\left[X|X_2\right]=\frac{5}{16}$
4. $P\left[X|X_1\right]=\frac{7}{16}$

Q.

If odds against an event be 7 : 9, find the probability of non-occurrence of this event.

Q. An instructor has a question bank consisting of $300$ easy True/False questions, $200$ difficult True/False questions, $500$ easy multiple choice questions and $400$ difficult multiple choice questions. If a question is selected at random from the question bank, then the probability that it will be an easy question given that it is a multiple choice question is:

1. $\frac{8}{14}$
2. $\frac{9}{14}$
3. $\frac{5}{14}$
4. $\frac{5}{9}$

Q.

Suppose that $5\%$ of men and $0.25\%$ of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Q.

Two dice are thrown. Find the odds in favour getting the sum

(i) 4

(ii) 5

(iii) What are the odd against getthing the sum 6?

Q. The probability that a two digit number selected at random will be a multiple of ${}^{\prime }{3}^{\prime }$ but not a multiple of ${}^{\prime }{5}^{\prime }$ is

1. $\frac{7}{15}$
2. $\frac{1}{3}$
3. $\frac{4}{15}$
4. $\frac{2}{5}$

Q.

Five cards are drawn successively with replacement from well- shuffled deck of 52 cards. What is the probability that

(i) all the five cards are spades?

(ii) only 3 cards are spades?

(iii) None is a spade?

Q. A point is selected at random inside an equilateral triangle whose side length is $3$ units. If the probability that its distance from any corner of the triangle is greater than $1$ unit is $1-\frac{2\pi }{k\sqrt{3}}$, then $k$ is

Q. A dice is thrown two times and the sum of the scores appearing on the dice is observed to be a multiple of $4$. Then the conditional probability that the score $4$ has appeared atleast once is

1. $\frac{1}{3}$
2. $\frac{1}{4}$
3. $\frac{1}{8}$
4. $\frac{1}{9}$

Q. A family has two children. The probability that both the children are boys given that atleast one of them is a boy is:

1. $\frac{1}{4}$
2. $\frac{2}{3}$
3. $\frac{3}{4}$
4. $\frac{1}{3}$

Q. A dice is thrown twice and the sum of numbers appearing is observed to be $6$. The conditional probability that the number $4$ has appeared atleast once is:

1. $\frac{2}{15}$
2. $\frac{3}{5}$
3. $\frac{2}{7}$
4. $\frac{2}{5}$

Q.

What are the odds in favour of getting a space if a card is drawn from a well-shuffled deck of cards? What are the odds in favour of getting a king?

Q. A box contains $10$ green, $20$ red, $30$ black and $40$ orange marble. One marble is drawn from the box at random. The probability that marble is neither green nor orange is:

1. $\frac{1}{2}$
2. $\frac{7}{10}$
3. $\frac{3}{5}$
4. $\frac{1}{3}$

Q. A question can be solved by $2$ methods. The probability of getting the correct answer using the method I and II are $\frac{1}{7}$ and $\frac{1}{8}$ respectively. If the problem was solved incorrectly then the probability that method I is

1. $\frac{47}{97}$
2. $\frac{48}{97}$
3. $\frac{47}{100}$
4. $\frac{49}{97}$

Q.

Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is

1. $\frac{14}{29}$

2. $\frac{16}{29}$

3. $\frac{15}{29}$

4. $\frac{10}{29}$

Q.

One of the two events must happen. Given that the chance of one is two -third of the other, find the odds in favour of the other.

Q. Two events $A$ and $B$ have the probabilities $0.25$ and $0.5$ respectively. The probability that both $A$ and $B$ occur simultaneously is 0.14. Then the probability of neither A nor B occurs is equal to:

1. $0.19$
2. $0.61$
3. $0.39$
4. $0.29$

Q. If $A,B,C$ are three events are such that $P\left(B\right)=\frac{3}{4},P(A\cap B\cap C^{\prime })=\frac{1}{3}$ and $P(A^{\prime }\cap B\cap C^{\prime })=\frac{1}{3},$ then $P(B\cap C)$ is equal to

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{1}{6}$
4. $\frac{1}{12}$

Q. $A$ post a letter to $B$ and inform him over the telephone. It is known that one letter out of $10$ letters does not reach its destination. If it is certain that $B$ will reply if he receives the letter and even if $B$ does not receive the letter, he replies with a probability of $\frac{1}{5}$ as he got the information that $A$ has written him a letter. If $X$ denotes the event that $B$ receives the letter and $Y$ denotes the event that $A$ gets a reply, then

1. $P(X\cap Y)=\frac{81}{100}$
2. $P\left(X|\overline{Y}\right)=\frac{45}{86}$
3. $A$ gets a reply, then the probability that $B$ receives the letter is $\frac{45}{46}$
4. $A$ gets a reply, then the probability that $B$ receives the letter is $\frac{5}{46}$

Q. Let $n_1$ and $n_2$ be the number of red and black balls, respectively, in box $I$. Let $n_3$ and $n_4$ be the number of red and black balls, respectively, in box $II$.

One of the two boxes, box $I$ and box $II$, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box $II$ is $\frac{1}{3}$, then the correct option(s) with the possible values of $n_1,n_2,n_3$ and $n_4$ is(are)

1. $n_1=3,n_2=3,n_3=5,n_4=15$
2. $n_1=3,n_2=6,n_3=10,n_4=50$
3. $n_1=8,n_2=6,n_3=5,n_4=20$
4. $n_1=6,n_2=12,n_3=5,n_4=20$

Q.

Five cards are drawn successively with replacement from well- shuffled deck of 52 cards. What is the probability that

(ii) only 3 cards are spades?

Q. The odds in favour of A solving a problem are 3 to 4 and the odds against B solving the same problem are 5 to 7. If they both try the problem, the probability that the problem is solved is:

1. $\frac{41}{84}$
2. $\frac{5}{21}$
3. $\frac{16}{21}$
4. $\frac{1}{4}$