Independent Events

Q.

State whether each of the following statements is true or false. Justify your answer.

(i) {2, 3, 4, 5} and (3, 6} are disjoint sets.

(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.

(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.

Q.

One mapping (function) is selected at random from all the mappings of set $A=\left\{1,2,3,..,n\right\}$ into itself. The probability that the mapping selected is one - one, is

1. $\frac{n!}{n^n-1}$

2. $\frac{n!}{n^n}$

3. $\frac{n!}{2n^n}$

4. None of these

Q. A bag contains 10 blue and 7 yellow balls. A ballis selected at random from the bag and replacedback into the bag. Again a ball is drawn from thebag. What is the probability that the first ball is ayellow ball and the second ball is a blue ball?

Q. (a) If A and B be mutually exclusive events associated with a random experiment such that P(A) = 0.4 and P(B) = 0.5, then find:
(i) P (A ∪ B)
(ii) $P(\overline{A}\cap \overline{B})$
(iii) P ($\overline{A}$ ∩ B)
(iv) P (A ∩ $\overline{B}$).

(b) A and B are two events such that P (A) = 0.54, P (B) = 0.69 and P (A ∩ B) = 0.35.
Find (i) P (A ∪ B)
(ii) $P(\overline{A}\cap \overline{B)}$
(iii) P (A ∩ $\overline{B}$)
(iv) P (B ∩ $\overline{A}$)

(c) Fill in the blanks in the following table:

	P (A)	P (B)	P (A ∩ B)	P (A ∪ B)
(i)	$\frac{1}{3}$	$\frac{1}{5}$	$\frac{1}{15}$	......
(ii)	0.35	....	0.25	0.6
(iii)	0.5	0.35	.....	0.7

Q. An unbiased coin is tossed. If the outcome is head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well shuffled pack of nine cards numbered $1,2,\cdots 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is :

1. $\frac{19}{36}$
2. $\frac{19}{72}$
3. $\frac{15}{72}$
4. $\frac{13}{36}$

Q. A fair coin is tossed three times. Consider the following events.
$E_1:$ Heads on the first toss.
$E_2:$ Tails on the second toss.
$E_3:$ Heads on the third toss.
$E_4:$ All three outcomes are same.
$E_5:$ Exactly one heads turns up.

Which of the following pairs/triples of events is (are) independent?

1. $E_1$ and $E_4$
2. $E_1$ and $E_5$
3. $E_1,E_2$ and $E_3$
4. $E_1,E_2$ and $E_4$

Q.

If $A$ and $B$ are two independent events such that$P(A\prime \cap B)=\frac{2}{15}$ and$P(A\cap B\prime )=\frac{1}{6}$, the $P\left(B\right)$ is

Q. If the odds in favour of winning a race by three horses are $1:4,1:5$ and $1:6$, find the probability that exactly one of these horses will win.

1. $\frac{37}{60}$
2. $\frac{39}{105}$
3. $\frac{41}{105}$
4. $\frac{51}{105}$

Q. Two dices are rolled one after the other. The probability that the number on the first is smaller than the number on the second is

1. $\frac{1}{2}$
2. $\frac{7}{18}$
3. $\frac{3}{4}$
4. $\frac{5}{12}$

Q. Anurn 5 red and 5 black balls .A ball is drawn at random its colour is noted and is returned to the urn moreover two additional balls of the colour drawn are put in the urn and then a ball is drawn at random .what is the probability that second ball is red?

Q.

For two events A and B, if
P(A)=P(A | B) =1/4 and P(B | A) =1/2, then

1. (a) A and B are independent
   

2. (b) A and B are mutually exclusive
   

3. (c) P(A' | B ') =3/4
   

4. (d) P(B' | A') =1/2

Q. If $A$ and $B$ are two independent events such that $P(A^{\prime }\cap B)=\frac{2}{15}$ and $P(A\cap B^{\prime })=\frac{1}{6},$ then $P\left(A\right)$ is

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

Q. A person draws a card from a pack, replaces it shuffles the pack, again draws a card, replace it and draws again. This process he does until he draw a heart card. The probability that he will have to make at least four draws is

1. $\frac{27}{256}$
2. $\frac{17}{256}$
3. $\frac{27}{64}$
4. $\frac{27}{16}$

Q. If $A$ and $B$ are two independent events such that $P\left(A\right)>0.5,$ $P\left(B\right)>0.5,$ $P(A\cap \overline{B})=\frac{3}{25},$ $P(\overline{A}\cap B)=\frac{8}{25},$ then the value of $P(A\cap B)$ is

1. $\frac{12}{25}$
2. $\frac{14}{25}$
3. $\frac{18}{25}$
4. $\frac{24}{25}$

Q. A box $B_1$ contains $1$ white ball, $3$ red balls, and $2$ black balls. Another box $B_2$ contains $2$ white balls, $3$ red balls, and $4$ black balls. A third box $B_3$ contains $3$ white balls, $4$ red balls, and $5$ black balls.
If 1 ball is drawn from each of the boxes $B_1,B_2$ and $B_3$ the probability that all $3$ drawn balls are of the same colour is

1. $\frac{82}{648}$
2. $\frac{90}{648}$
3. $\frac{558}{648}$
4. $\frac{566}{648}$

Q. In shuffling a pack of 52 playing cards, four are accidently dropped; find the chance that the missing cards should be one from each suit.

Q. A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are, respectively, $p$, $q$ and $1/2$. if the probability that the student is successful is $1/2$, then $p(1+q)=$

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

Q. A bag contains 'a' white and 'b' black balls. Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw. A begins the game. If the probability of A winning ( that is
drawing a white ball) is twice the probability of B winning, then the ratio a : b is equal to

1. 1 : 2
2. 2 : 1
3. 1 : 1
4. 1 : 3

Q. The probability that a teacher will give an unannounced test during any class meeting is $\frac{1}{5}$.If a student is absent twice, the probability that he will miss atleast one test, is

1. $\frac{7}{25}$
2. $\frac{9}{25}$
3. $\frac{16}{25}$
4. $\frac{24}{25}$

Q.

The conditional $(p\wedge q)\Rightarrow p$ is ___.

1. A tautology

2. A fallacy

3. Neither a tautology nor a fallacy

4. A contrapositive

Q. Two cards are drawn from the standard deck of 52 playing cards without replacement. Find the probability of getting first card as queen and second as an ace.

1. $\frac{1}{169}$
2. $\frac{4}{663}$
3. $\frac{1}{13}$
4. $\frac{2}{13}$

Q.

If A = {x: x is a natural number}, B ={x: x is an even natural number}

C = {x: x is an odd natural number} and D = {x: x is a prime number}, find

(i) A ∩ B

(ii) A ∩ C

(iii) A ∩ D

(iv) B ∩ C

(v) B ∩ D

(vi) C ∩ D

Q. The probability that the ${13}^{th}$ day of a randomly chosen month is a Friday, is

1. $\frac{1}{12}$
2. $\frac{2}{3}$
3. $\frac{1}{84}$
4. $None$

Q. A man alternately tosses a coin and throws a die beginning with coin. The probability that he gets a head in the coin before he gets 5 or 6 on the die is

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

Q. If two switches S1 and S2 have respectively 90% and 80% chances of working, then the value of 100 X probability that the circuit will work = ___

Q. If the probability for A to fail in an examination is 0.2 and that for B is 0.3, the probability that either A or B fails, is

1. 0.38
2. 0.44
3. 0.5
4. 0.94