Axiomatic Approach

Q.

An urn contains 9 red, 7 white and 4 black balls. If two balls are drawn at random find the probability that

(i) both the balls are red.

(ii) one is white and other is red.

(iii) the balls are of same Colour.

Or

A box contains 10 bulbs, of which just three are defective. If at random a sample of five bulbs is drawn, find the probabilities that the sample contains

(i) exactly one defective bulb.

(ii) exactly two defective bulbs.

(iii) no defective bulbs.

Q.

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

Q.

In a lottery, a person chooses six different numbers at random from 1 to 20. If these six numbers match with the six numbers already fixed by the lottery committee he wins the prize. What is the probability of winning the prize in the game ?

Q.

The probability that a person will get an electrification contract is $\left(\frac{2}{5}\right)$ and the probability that he will not get a plumbing contract is $\frac{4}{7}.$ If the probability of getting at least one contract is $\left(\frac{2}{3}\right)$, what is the probability that he will get both ?

Q.

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many ways these

(i) four cards are of the same suit?

(ii) four cards are belongs to four different suits?

(iii) four cards are of the same colour?

Q.

A natural number is chosen at random from among the first 500. What is the probability that the number so chosen is divisible by 3 or 5 ?

Q.

If $\frac{2}{11}$ is the probability of an event, what is
the probability of the event not A'

Q.

$AandB$ are two events such that $P\left(A\right)=0.8,P\left(B\right)=0.6,andP(A\cap B)=0.5$, then the value of $P\left(\frac{A}{B}\right)$ is

1. $\frac{5}{6}$

2. $\frac{5}{8}$

3. $\frac{9}{10}$

4. None of these

Q.

In a single throw of two dice, determine the probability of not getting same number on the two dice.

Q.

A letter is taken at random from the letters of the word 'STATISTICS' and another letter is taken at random from the letters of the word 'ASSISTANT'. The probability that they are the same letter is

1. 1/45
   

2. 13/90
   

3. 19/90
   

4. 5/18

Q.

The probability that at least one of the events $E_1$ and $E_2$ occurs is 0.6 If the probability of the simultaneous occurrence of $E_1$ and $E_2$ is 0.2, find $P\left({\overline{E}}_1\right)+P\left({\overline{E}}_2\right).$

Q.

The coefficients $a,b\text{and}c$ of the quadratic equation $a{x}^2+bx+c=0$ is obtained by throwing a dice three times. The probability that this equation has equal roots is

1. $\frac{1}{54}$

2. $\frac{1}{72}$

3. $\frac{1}{36}$

4. $\frac{5}{216}$

Q.

A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective, if a person takes out 2 at random, what is the probability that either both are apples or both are good?

Q.

In a certain lottery 10, 000 tickets are sold and ten equal prizes are awared. What is the probability of not getting a prize if you buy (a) one ticket (b) two tickets (c) 10 tickets.

Q.

If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with replacement, determine the probability that the roots of the equation $x^2+px+q=0$ are real.___

Q.

The probability that a patient visiting a dentist will have a tooth extracted is 0.06, the probability that he will have a cavity filled is 0.2, and the probability that he will have a tooth extracted or a cavity filled is 0.23. What is the probability that he will have a tooth extracted as well as a cavity filled ?

Q.

In how many ways can 19 identical books on English and 17 identical books on Hindi be placed in a row on a shelf so that two books on Hindi may not be together ?

Q.

Which of the following cannot be valid assignment of probabilities for outcomes of sample space
S = $$\left\{\begin{array}{ccccccc}{w}_1,& w_2,& w_3,& w_4,& w_5,& w_6,& w_7,\end{array}\right\}$$

$$\begin{array}{cccccccc}& w_1& w_2& w_3& w_4& w_5& w_6& w_7\\ \left(a\right)& 0.1& 0.1& 0.05& 0.03& 0.01& 0.2& 0.6\\ \left(b\right)& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}& \frac{1}{7}\\ \left(c\right)& 0.1& 0.2& 0.3& 0.4& 0.5& 0.6& 0.7\\ \left(d\right)& -0.1& 0.2& 0.3& 0.4& -0.2& 0.1& 0.3\\ \left(e\right)& \frac{1}{14}& \frac{2}{14}& \frac{3}{14}& \frac{4}{14}& \frac{5}{14}& \frac{6}{14}& \frac{15}{14}\end{array}$$

Q.

If a letter is chosen at random from the English alphabet, find probability that the letter chosen is

(i) a vowel, and

(ii) a consonant

Q.

Determine the number of 4 card combinations out of a deck of 52 cards if there is no ace in each combination.

1. 

2. 

3. 

4. 

Q.

In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present?

1. 

2. 

3. 

4. 

5. 

Q.

What is the probability that an ordinary year has 53 Tuesdays ?

Q.

One urn contains two black balls (labelled $B_1$ and $B_2$). Suppose, the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then, a second ball is chosen at random from the same urn without replacing the first ball.

(i) Write the sample space showing all possible outcomes.

(ii) What is the probability that two black balls are chosen ?

(iii) What is the probability that two balls of opposite colour are chosen ?

Q. For any two independent events $E_1$ and $E_2$ in a space $S$, $P\left[\right(E_1\cup E_2)\cap (E_1\cap E_2\left)\right]$ is equal to

1. $\le \frac{1}{4}$
2. $>\frac{1}{4}$
3. $\ge \frac{1}{2}$
4. $>\frac{1}{2}$

Q. The probability that A speaks truth is $\frac{4}{5}$, while this probability for B is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact is

1. 3/23
2. 1/20
3. 1/5
4. 14/40

Q.

A, B and C are events such that $P\left(A\right)=0.3,P\left(B\right)=0.4,P\left(C\right)=0.8,P(A\cap B)=0.08,P(A\cap C)=0.28$and $A\cup B\cup C)\ge 0.75$ show that $P(B\cap C)$ lies in the interval [0.23, 0.48]

Or

An integer is chosen random from 1 to 50, what is the probability that the integer chosen, is a multiple of 2 or 3 or 10?

Q.

A die is thrown. Describe the following events.
(i) A: a number less than 7
(ii) B: a number greater than 7
(iii) C: a multiple of 3
(iv) D: a number less than 4
(v) E: an even number greater than 4
(vi) F: a number not less than 3
Also find $A\cup B,A\cap B,B\cup C,E\cap F,D\cap E,A-C,D-E,E\cap F^{\prime },F^{\prime }$

Q. The number of ways that a volley ball 6 can be selected out of 10 players so that 2 particular players are excluded is

1. 56
2. 55
3. 27
4. 28

Q.

Cards are drawn one by one at random from a well shuffled full pack of 52 cards until two aces are obtained for the first time. If N is the number of cards required to be drawn, then $P_{r}N=n$ where $2\le n\le 50,$ is

1. 
2. 
3. 
4. 

Q.

In how many ways the sum of upper faces of four distinct dies can be six.

1. 10

2. 20

3. 

4.