Probability · NDA previous year questions

If A and B are two events such that $\mathrm{P}(\mathrm{A})=\frac{3}{4}$ and $\mathrm{P}(\mathrm{B})=\frac{5}{8}$, then consider the following statements: 1. The minimum value of $\mathrm{P}(\mathrm{A}\cup \mathrm{B})$ is $\frac{3}{4}$. 2. The maximum value of $\mathrm{P}(\mathrm{A}\cap \mathrm{B})$ is $\frac{5}{8}$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If three dice are rolled under the condition that no two dice show the same face, then what is the probability that one of the faces is having the number 6 ?

1. A.$\frac{5}{6}$
2. B.$\frac{5}{9}$
3. C.$\frac{1}{2}$
4. D.$\frac{5}{12}$

If $P(A\cup B)=\frac{5}{6}$, $P(A\cap B)=\frac{1}{3}$ and $P(\text{not }A)=\frac{1}{2}$, then which one of the following is not correct?

1. A.$P(B)=\frac{2}{3}$
2. B.$P(A\cap B)=P(A)P(B)$
3. C.$P(A\cup B)>P(A)+P(B)$
4. D.$P(\text{not }A\text{ and not }B)=P(\text{not }A)P(\text{not }B)$

What is the probability that February of a leap year selected at random, will have five Sundays?

1. A.$\frac{1}{5}$
2. B.$\frac{1}{7}$
3. C.$\frac{2}{7}$
4. D.1

A husband and wife appear in an interview for two vacancies for the same post. The probability of the husband's selection is $\frac{1}{7}$ and that of the wife's selection is $\frac{1}{5}$. If the events are independent, then the probability of which one of the following is $\frac{11}{35}$?

1. A.At least one of them will be selected
2. B.Only one of them will be selected
3. C.None of them will be selected
4. D.Both of them will be selected

A dealer has a stock of 15 gold coins out of which 6 are counterfeit. A person randomly picks 4 of the 15 gold coins. What is the probability that all the coins picked will be counterfeit?

1. A.$\frac{1}{91}$
2. B.$\frac{4}{91}$
3. C.$\frac{6}{91}$
4. D.$\frac{15}{91}$

A committee of 3 is to be formed from a group of 2 boys and 2 girls. What is the probability that the committee consists of 2 boys and 1 girl?

1. A.$\frac{2}{3}$
2. B.$\frac{1}{4}$
3. C.$\frac{3}{4}$
4. D.$\frac{1}{2}$

In a lottery of 10 tickets numbered 1 to 10, two tickets are drawn simultaneously. What is the probability that both the tickets drawn have prime numbers?

1. A.$\frac{1}{15}$
2. B.$\frac{1}{2}$
3. C.$\frac{2}{15}$
4. D.$\frac{1}{5}$

Consider a random variable $X$ which follows Binomial distribution with parameters $n=10$ and $p=\frac{1}{5}$. Then $Y=10-X$ follows Binomial distribution with parameters $n$ and $p$ respectively given by

1. A.$5$, $\frac{1}{5}$
2. B.$5$, $\frac{2}{5}$
3. C.$10$, $\frac{3}{5}$
4. D.$10$, $\frac{4}{5}$

If $A$ and $B$ are two events such that $P(A)=0.6$, $P(B)=0.5$ and $P(A\cap B)=0.4$, then consider the following statements: 1. $P(\bar{A}\cup B)=0.9$. 2. $P(\bar{B}\mid \bar{A})=0.6$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Three cooks $X$, $Y$ and $Z$ bake a special kind of cake, and with respective probabilities 0.02, 0.03 and 0.05, it fails to rise. In the restaurant where they work, $X$ bakes 50%, $Y$ bakes 30% and $Z$ bakes 20% of cakes. What is the proportion of failures caused by $X$?

1. A.$\frac{9}{29}$
2. B.$\frac{10}{29}$
3. C.$\frac{19}{29}$
4. D.$\frac{28}{29}$

A bag contains 20 books out of which 5 are defective. If 3 of the books are selected at random and removed from the bag in succession without replacement, then what is the probability that all three books are defective?

1. A.0.009
2. B.0.016
3. C.0.026
4. D.0.047

A coin is biased so that heads comes up thrice as likely as tails. For three independent tosses of a coin, what is the probability of getting at most two tails?

1. A.0.16
2. B.0.48
3. C.0.58
4. D.0.98

If a coin is tossed till the first head appears, then what will be the sample space?

1. A.{H}
2. B.{TH}
3. C.{T, HT, HHT, HHHT, ................}
4. D.{H, TH, TTH, TTTH, ................}

If a fair die is rolled 4 times, then what is the probability that there are exactly 2 sixes?

1. A.$\frac{5}{216}$
2. B.$\frac{25}{216}$
3. C.$\frac{125}{216}$
4. D.$\frac{175}{216}$

Consider the following statements : 1. If $A$ and $B$ are mutually exclusive events, then it is possible that $P(A)=P(B)=0.6$. 2. If $A$ and $B$ are any two events such that $P(A|B)=1$, then $P(\bar{B}|\bar{A})=1$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

There are 3 coins in a box. One is a two-headed coin; another is a fair coin; and third is biased coin that comes up heads 75% of time. When one of the three coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

1. A.$\frac{2}{9}$
2. B.$\frac{1}{3}$
3. C.$\frac{4}{9}$
4. D.$\frac{5}{9}$

If 5 of a Company's 10 delivery trucks do not meet emission standards and 3 of them are chosen for inspection, then what is the probability that none of the trucks chosen will meet emission standards?

1. A.$\frac{1}{8}$
2. B.$\frac{3}{8}$
3. C.$\frac{1}{12}$
4. D.$\frac{1}{4}$

Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on them is a prime number?

1. A.$\frac{5}{12}$
2. B.$\frac{1}{2}$
3. C.$\frac{7}{12}$
4. D.$\frac{2}{3}$

What is the value of $$2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\dots \infty }}}?$$

1. A.$\sqrt{2}-1$
2. B.$\sqrt{2}+1$
3. C.3
4. D.4

Let the correlation coefficient between X and Y be 0.6. Random variables Z and W are defined as $Z=X+5$ and $W=\frac{Y}{3}$. What is the correlation coefficient between Z and W?

1. A.0.1
2. B.0.2
3. C.0.36
4. D.0.6

From a deck of cards, cards are taken out with replacement. What is the probability that the fourteenth card taken out is an ace?

1. A.$\frac{1}{51}$
2. B.$\frac{4}{51}$
3. C.$\frac{1}{52}$
4. D.$\frac{1}{13}$

If A and B are two events such that P(A) = 0.5, P(B) = 0.6 and $P(A\cap B)=0.4$, then what is $P(\overline{A\cup B})$ equal to?

1. A.0.9
2. B.0.7
3. C.0.5
4. D.0.3

A problem is given to three students A, B and C whose probabilities of solving the problem are $\frac{1}{2}$, $\frac{3}{4}$ and $\frac{1}{4}$ respectively. What is the probability that the problem will be solved if they all solve the problem independently?

1. A.$\frac{29}{32}$
2. B.$\frac{27}{32}$
3. C.$\frac{25}{32}$
4. D.$\frac{23}{32}$

A pair of fair dice is rolled. What is the probability that the second dice lands on a higher value than does the first?

1. A.$\frac{1}{4}$
2. B.$\frac{1}{6}$
3. C.$\frac{5}{12}$
4. D.$\frac{5}{18}$