Independent and Dependent Events

Q. Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

1. $\frac{16}{5525}$
2. $\frac{64}{2197}$
3. $\frac{3}{13}$
4. $\frac{8}{16575}$

Q.

From a pack of playing cards three cards are drawn simultaneously. The probability that these are one king, one queen and one jack is

1. $\frac{64}{5525}$

2. $\frac{16}{5525}$

3. $\frac{128}{5525}$

4. $\frac{64}{625}$

Q.

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colors is

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

2. $\frac{2}{7}$

3. $\frac{1}{21}$

4. $\frac{2}{23}$

Q. Ram and Ramesh appeared in an interview for two vacancies in the same department. The probabilities of Ram's selection is 1/6 and that of Ramesh is 1/8. What is probability that only one of them will be selected.

1. 47/48
2. 1/4
3. 13/48
4. 35/48

Q. Consider a e with the property that the probability of a face with n dots showing up is proportional to n. The probability of the face with three dots showing up is .

1. 0.142

Q. In a population of N families, 50% of the families have three children, 30% of the families have two children and the remaining families have one child. What is the probability that a randomly picked child belongs to a family with two children?

1. $\frac{3}{23}$
2. $\frac{6}{23}$
3. $\frac{3}{10}$
4. $\frac{3}{5}$

Q. P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}.$ The probability that P applies for the job given that Q applies for the job is $\frac{1}{2},$ and the probability that Q applies for the job given that P applies for the job is $\frac{1}{3}.$ Then the probability that P does not apply for the job given that Q does not apply for the job is

1. $\frac{4}{5}$
2. $\frac{5}{6}$
3. $\frac{7}{8}$
4. $\frac{11}{12}$

Q. Given Set A = {2, 3, 4, 5} and Set B = {11, 12, 13, 14, 15}, two numb ers are randomly selected one from each set. What is the probability that the sum of the two numbers equals 16?

1. 0.20
2. 0.25
3. 0.30
4. 0.33

Q. The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that it will rain either today or tomorrow is 0.7. What is the probability that it will rain today and tomorrow?

1. 0.3
2. 0.25
3. 0.35
4. 0.4

Q. Let P(E) denote the probability of an event E. Given $P\left(A\right)=1,P\left(B\right)=\frac{1}{2}$, the values of P(A/B) and P(B/A) respectively are

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

Q. A cab was involved in a hit and run accident at night. You are given the following data about the cabs in the city and the accident.
(i) 85% of cabs in the city are green and the remaining cabs are blue.
(ii) A witness identified the cab involved in the accident as blue.
(iii) It is known that a witness can correctly identify the cab colour only 80% of the time.

Which of the following options is closest to the probability that the accident was caused by a blue cab?

1. 12%
2. 15%
3. 41%
4. 80%

Q. Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probability 1/2, 1/4 and 1/4 respectively. What is the conditional probability P(X + Y = 2/X - y = 0) ?

1. 0
2. 1/16
3. 1/6
4. 1

Q.

A box contains $10$ identical electronic components of which $4$ are defective. If $3$ components are selected at random from the box in succession, without replacing the units already drawn, what is the probability that two of the selected components are defective?

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

2. $\frac{5}{24}$

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

4. $\frac{1}{40}$

Q. A box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set contains one red ball and two black balls is

1. $\frac{1}{20}$
2. $\frac{1}{12}$
3. $\frac{3}{10}$
4. $\frac{1}{2}$

Q.

The input X to the Binary Symmetric Channel (BSC) shown in the figure '1' with probability 0.8.
The crossover probability is 1/7. If the received bit Y = 0, . the conditional probability that '1' was transmitted is .

1. 0.4

Q. An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the second draw is

1. $\frac{1}{2}$
2. $\frac{4}{9}$
3. $$\frac{5}{9}$$
4. $$\frac{6}{9}$$

Q. A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective?

1. $\frac{1}{5}$
2. $\frac{1}{25}$
3. $\frac{20}{99}$
4. $\frac{19}{495}$

Q. There are 25 calculators in a box. Two of them are defective. suppose 5 calculators are randomly picked for inspection (i.e., each has the same chance of being selected), what is the probability that only one of the defective calculators will be include in the inspection?

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

Q. Suppose A and B are two independent events with probabilities $P\left(A\right)\ne 0\text{and}P\left(B\right)\ne 0.$ $\text{Let}\overline{A}\text{and}\overline{B}$ be their complements. Which one of the following statements is FALSE?

1. $P(A\cap B)=P\left(A\right)P\left(B\right)$
2. $P\left(A/B\right)=P\left(A\right)$
3. $P(A\cup B)=P\left(A\right)+P\left(B\right)$
4. $P(\overline{A}\cap \overline{B})=P\left(\overline{A}\right).P\left(\overline{B}\right)$

Q. In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is

1. 0.667

Q. Let S be a sample space and two mutually exclusive events A and B such that $A\cup B=S.$ If P(.) denotes the probability of the event, the maximum value of P(A)P(B) is .

1. 0.25

Q. Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specifications are 0.8, 0.7 and 0.5. Each vendor supplies one component. The probability that out of total three components supplied by the vendors at least one will meet the design specifications is .

1. 0.97

Q. A hydraulic structure has four gates which operate independently. The probability of failure of each gate is 0.2. Given that gate 1 has failed, the probability that both gates 2 and 3 will fail is

1. 0.240
2. 0.200
3. 0.040
4. 0.008

Q. If P and Q are two random events, then the following is TRUE

1. Independence of P and Q implies that probability $(P\cap Q)=0$
2. Probability $(P\cup Q)\ge$ Probability (P) + Probability (Q)
3. If P and Q are mutually exclusive, then they must be independent
4. Probability $(P\cap Q)\le$ Probability (P)

Q. $E_{1}and{E}_2$ are events in a probability space satisfying the following constraints $P\left(E_1\right)=P\left(E_2\right);P(E_1\cup E_2)=1;E_{1}and{E}_2$ are independent then $P\left(E_1\right)=$

1. 0
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. 1

Q. Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes is equally likely at both places. He can afford up to 13 minutes of total waiting time if he is to arrive at D on time. What is the probability that Manish will arrive late at D?

1. $\frac{8}{13}$
2. $\frac{13}{64}$
3. $\frac{119}{128}$
4. $\frac{9}{128}$

Q. A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occur in three tosses. Based on the above information which one of the following statements is TRUE?

1. X and Y are not independent
2. Y and Z are dependent
3. Y and Z are independent
4. X and Z are independent

Q. In a class of 200 students, 125 students have taken programming language course, 85 students have taken data structures course, 65 students have taken computer organization course, 50 students have taken both programming languages and data structures, 35 students have taken both programming languages and computer organization, 30 students have taken both data structures and computer organization, 15 students have taken all the three courses. How many students have not taken any of the three courses?

1. 15
2. 20
3. 25
4. 35

Q. Two fair dice are rolled and the sum r of the numbers turned up is considered

1. $Pr(r>6)=\frac{1}{6}$
2. $Pr\left(r/3isaninteger\right)=\frac{5}{6}$
3. $Pr(r=8/(r/4\left)isaninteger\right)=\frac{5}{9}$
4. $Pr(r=6/(r/5\left)isaninteger\right)=\frac{1}{18}$

Q. In any given year, the probability of an earthquake greater than magnitude 6 occuring in the Garhwal Himalayas is 0.04. The average time between successive occurance of such earthquake in years.

1. 25