Characteristics of Axiomatic Approach

Q.

What are the $4$ Laws of Probability?

Q.

An unbiased die with faces marked $1,2,3,4,5$ and $6$ is rolled four times. Out of four values obtained the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

1. $\frac{16}{81}$

2. $\frac{1}{81}$

3. $\frac{80}{81}$

4. $\frac{65}{81}$

Q. A letter is chosen at random from the word $“STATISTICS”$. The probability of getting a vowel is

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

Q. 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 $\frac{1}{2}$
2. Exactly equal to $\frac{1}{2}$
3. Greater than $\frac{1}{8}$ but less than $\frac{1}{4}$
4. Greater than $\frac{1}{4}$ but less than $\frac{1}{2}$

Q.

A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of

(i) 5 successes? (ii) at least 5 successes?

(iii) at most 5 successes?

Q.

Out of a pack of 52 cards one is lost, from the remainder of the pack , two cards are drawn and are found to be spades .Find the chance that the missing card is a spade ?

1. 11/50

2. 11/49

3. 10/49

4. 10/50

Q.

Given that E and F are events such that P (E) = 0.6 P (F) = 0.3 and
P$(E\cap F)=0.2,find\left(\frac{E}{F}\right)andP\left(\frac{F}{E}\right).$

Q.

On a multiple choice examination with three possible answer for each of the five questions, what is the probability that a candidate would get four or more correct answer just by guessing ?

Q.

What are the two basic laws of probability?

Q.

Let $S$ be the sample space of the random experiment of throwing simultaneously two unbiased dice with six faces numbered $1$ to $6$ and Given, $E_k=\left\{\left(a,b\right)\in S:ab=k\right\}$ for $k\ge 1$. If $P_k=P\left(E_k\right)$ for $k\ge 1$, Then the correct among the following.

1. $P_1<P_{30}<P_4<P_6$

2. $P_{36}<P_6<P_2<P_4$

3. $P_1<P_{11}<P_4<P_6$

4. $P_{36}<P_{11}<P_6<P_4$

Q.

A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Q. A forecast is to be made of the results of five cricket matches, each of which can be a win or a draw or a loss for Indian team. Let,
$p=$ number of forecasts with exactly $1$ error
$q=$ number of forecasts with exactly $3$ error
$r=$ number of forecasts with all five error
Then the correct statement(s) is/are

1. $2q=5r$
2. $8p=q$
3. $8p=5r$
4. $2(p+r)>q$

Q.

If odds against solving a question by three students are $2:1,5:2$ and $5:3$ respectively, then the probability that the question is solved only by one student is

1. $\frac{32}{56}$

2. $\frac{24}{56}$

3. $\frac{25}{56}$

4. None of these

Q.

A die is tossed twice. The probability of getting 1, 2, 3 or 4 on the first toss and 4, 5, or 6 on the second toss is:

1. 1/2

2. 1/3

3. 1/9

4. 2/5

Q.

Let $A$ and $B$ be two independent events such that $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{1}{6}$. Then, which of the following is TRUE?

1. $P\left(\frac{A}{AUB}\right)=\frac{1}{4}$

2. $P\left(\frac{A}{B}\right)=\frac{1}{3}$

3. $P\left(\frac{A}{B}\right)=\frac{2}{3}$

4. $P\left(\frac{A}{B}\right)=\frac{1}{3}$

Q. Let $P\left(E\right)$ denote the probability of occurrence of event $E.$ If $A$ and $B$ are two events associated with an experiment such that $\left|\begin{array}{ccc}P\left(A\right)-1& P\left(B\right)& -P(A\cap B)\\ P\left(B\right)& -P(A\cap B)& P\left(A\right)-1\\ -P(A\cap B)& P\left(A\right)-1& P\left(B\right)\end{array}\right|=0,$ then

1. $P(A\cup B)=1$
2. $P\left(A|B\right)=\frac{P\left(A\right)+P\left(B\right)-1}{P\left(B\right)}$
3. If $A\subset B,$ then $P\left(B\right)=P(A\cup B)$
4. If $A\subset B,$ then $P\left(A\right)=1$

Q. A die is rolled. The probability of getting a positive integer less than 7, is

Q.

A coin is tossed twice, what is the probability that at least one tail occurs?

Q. Mark the correct alternative in the following question:

A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number of the die and a spade card is

$\left(\mathrm{a}\right)\frac{1}{2}\left(\mathrm{b}\right)\frac{1}{4}\left(\mathrm{c}\right)\frac{1}{8}\left(\mathrm{d}\right)\frac{3}{4}$

Q. Rural and urban students are equally likely to get admission in a college. If $100$ students get admission, then the probability that more rural students get admission than urban students is

1. ${\left(\frac{1}{2}\right)}^{100}{}^{100}{P}_{50}$
2. ${\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50}$
3. $1-{\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50}$
4. $\frac{1}{2}(1-{\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50})$

Q. The probability of happening of an event $A$ is $0.5$ and that of $B$ is $0.3$. If $A$ and $B$ are mutually exclusive events, then the probability of neither $A$ nor $B$ is

Q. An unbiased die with face marked 1, 2, 3, 4, 5, 6 is rolled four times. Out of 4 face values obtained, find the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5.

Q. Let A and B be two independent events such that $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{1}{4}$
$\begin{array}{cccc}& ListI& & ListII\\ P.& P(A\cup B)isequalto& 1.& \frac{1}{2}\\ Q.& P\left(\frac{A}{A\cup B}\right)isequalto& 2.& \frac{2}{3}\\ R.& P\left(\frac{B}{A^{\prime }\cap B^{\prime }}\right)isequalto& 2.& \frac{1}{3}\\ S.& P\left(\frac{A}{B}\right)isequalto& 4.& 0\end{array}$

1. P-2; Q-1; R-4; S-3
2. P-2; Q-3; R-1; S-4
3. P-1; Q-3; R-2; S-4
4. P-1; Q-2; R-4; S-3

Q.

Out of a pack of 52 cards one is lost, from the remainder of the pack , two cards are drawn and are found to be spades .Find the chance that the missing card is a spade ?

1. $\frac{11}{50}$

2. $\frac{11}{49}$

3. $\frac{10}{49}$

4. $\frac{10}{50}$

Q. If getting 5 or 6 in a throw of an unbiased die is a success and the random variable X denotes the number of successes in six throws of the die, find P (X ≥ 4).

Q. Let A and B be two events such that $P\left(A\right)=\frac{1}{3},P\left(\frac{A}{B}\right)=\frac{1}{2}$ and $P\left(\frac{B}{A}\right)=\frac{2}{5}$
$P\left(\frac{\overline{A}}{B}\right)=$

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

Q. Two cards are drawn from a well shuffled pack of 52 cards without replacement. The probability of drawing a queen and a jack is :

1. $\frac{4}{663}$
2. $\frac{8}{663}$
3. $\frac{16}{663}$
4. $\frac{2}{663}$

Q. A student appears for tests $I,II$ and $III$. The student is successful if he passes in tests $I,II$ or $I,III$. The probabilities of the student passing in tests $I,II$ and $III$ are respectively $p,q$ and $\frac{1}{2}$ . If the probability of the student to be successful is $\frac{1}{2}$. Then

1. $p(1+q)=1$
2. $\frac{1}{p}+\frac{1}{q}=1$
3. $q(1+p)=1$
4. $pq=1$

Q. A card is drawn at random from a well-shuffled pack of $52$ playing cards. Find the probability of the events that the card drawn is :
$\left(i\right)$a black card.

Q. If X and Y are two events such that $P\left(X|Y\right)=\frac{1}{2},P\left(Y|X\right)=\frac{1}{3}andP(X\cap Y)=\frac{1}{6}.$ Then, which of the following is/are correct ?

1. $P(X\cup Y)=\frac{2}{3}$
2. X and Y are independent
3. X and Y are not independent
4. $P(X^C\cap Y)=\frac{1}{3}$