Independent Events

Q.

Two dice are thrown together. If the numbers appearing on the two dice are different, then what is the probability that the sum is $6$ $?$

1. $\frac{5}{36}$

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

3. $\frac{2}{15}$

4. None of these

Q.

What Is The Probability That An Ordinary Year Has $53$ Sundays?

Q.

A basket contains $5$ apples and $7$ oranges and another basket contains $4$ apples and $8$ oranges. One fruit is picked out from each basket. Find the probability that the fruits are both apples or both oranges

1. $\frac{24}{144}$

2. $\frac{56}{144}$

3. $\frac{68}{144}$

4. $\frac{76}{144}$

Q.

A committee has to be made of $5$ members from $6$ men and $4$ women.

The probability that at least one woman is present in the committee is

1. $\frac{1}{42}$

2. $\frac{41}{42}$

3. $\frac{2}{63}$

4. $\frac{1}{7}$

Q.

The probability of India winning a test match against South Africa is $\frac{1}{2}$ assuming independence from match to match played. The probability that in a $5$ match series India’s second win occurs at the $3^{rd}$ test is

1. $\frac{1}{8}$

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

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

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

Q. Two independent events having nonzero probabilities of occurrence can not be mutually exclusive.

1. True
2. False

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. Which of the following relations hold true for two independent events A and B?

1. $P(A\cup B)=P\left(A\right)+P\left(B\right)$
2. $P(A\cap B)=0$
3. $P(A\cup B)=P\left(A\right).P\left(B\right)$
4. $P(A\cap B)=P\left(A\right).P\left(B\right)$

Q.

A person can kill a bird with probability$3/4$. He tries $5$times. What is the probability that he may not kill the bird

1. $243/1024$

2. $781/1024$

3. $1/1024$

4. $1023/1024$

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. From a bag containing 9 distinct white and 9 distinct black, 9 balls are drawn at random one by one, the drawn balls being replaced each time. The probability that at least four balls of each colour is in the draw, is

1. A little less than $\frac{1}{2}$
2. a little greater than $\frac{1}{2}$
3. $\frac{1}{2}$
4. $\frac{1}{3}$

Q. A bag contains 6 red and 4 blue balls (all are different). A fair die is rolled and number of balls equals to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red is

1. $\frac{1}{5}$
2. $\frac{1}{30}$
3. $\frac{127}{630}$
4. $\frac{1}{6}$

Q.

Let two fair six-faced dice A and B be thrown simulatneously. If $E_1$ is the event that die A shows up four, $E_2$ is the event that die B shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is/are true ?

1. $E_{1}and{E}_2$ are independent

2. $E_{2}and{E}_3$ are independent

3. $E_{1}and{E}_3$ are independent

4. $E_1,E_{2}and{E}_3$ are independent

Q. Suppose you have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. You are going to pull out one marble, record its color, put it back in the box and draw another marble. What is the probability of pulling out a red marble followed by a blue marble?

1. $\frac{8}{81}$
2. $\frac{2}{27}$
3. $\frac{1}{27}$
4. $\frac{1}{9}$

Q.

A box contains $20$ cards, numbered from $1$ to $20$. A card is drawn from the box at random. Find the probability that the number on the drawn card is even, prime, and a multiple of $3.$

Q. If A and B are independent events such that
$P\left(A\right)=p,P\left(B\right)=2p$ and
$P\left(\text{Exactly one of A and B}\right)=\frac{5}{9}$
then $p=$

1. $\frac{1}{3}$
2. $\frac{5}{6}$
3. $\frac{5}{12}$
4. $\frac{9}{11}$

Q.

If $A_1,A_2,.....,A_n$ are n independent events such that $P\left(A_i\right)=\frac{1}{i+1},i=1,2,....,n$. The probability that none of $A_1,A_2,....A_n$ occurs is

1. $\frac{n}{n+1}$

2. $\frac{1}{n+1}$

3. more than $\frac{1}{n}$

4. None of these

Q. Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3},$ respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2,$ respectively, after two games.

$P(X=Y)$ is

1. $\frac{11}{36}$
2. $\frac{1}{3}$
3. $\frac{13}{36}$
4. $\frac{1}{2}$

Q. A bag contains 6 red and 4 blue balls (all are different). A fair die is rolled and number of balls equals to that appearing on the die is chosen from the bag at random. The probability that all the balls selected are red is

1. $\frac{127}{630}$
2. $\frac{1}{5}$
3. $\frac{1}{30}$
4. $\frac{1}{6}$

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. For two independent events A and B, if P(A) = 0.5 and P(B) = 0.3 , then the value of
$100P(A\cup B)$ = ___

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. A bag contains (2n + 1) coins. It is known that n of these coins have a head on both sides whereas remaining coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is 31/42, then value of n be

1. 10
2. 11
3. 12
4. 13

Q. Let ‘A’, ‘B’ and ‘C’ be three independent events with $P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{2}$ and $P\left(C\right)=\frac{1}{4}$. The probability of exactly 2 of these events occurring, is equal to:

1. $\frac{1}{4}$
2. $\frac{7}{24}$
3. $\frac{3}{4}$
4. $\frac{17}{24}$

Q. Let A, B , and C be three independent events with $P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{2},andP\left(C\right)=\frac{1}{4}.$ . The probability of exactly 2 of these events occurring, is equal to

1. $\frac{1}{4}$
2. $\frac{7}{24}$
3. $\frac{3}{4}$
4. $\frac{17}{24}$

Q.

A bag contains $2n+1$ coins. It is known that $n$ of these coins have a head on both sides, whereas the remaining $(n+1)$ coins are fair. A coin is picked up at random from the bag and tossed. If the probability that the toss results in a head is $\frac{31}{42}$ then $n$ is equal to:

1. $10$

2. $11$

3. $12$

4. $13$

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. $P\left(A|\right(A\cup B\left)\right)=\frac{1}{4}$
2. $P\left(A|B^{\prime }\right)=\frac{1}{3}$
3. $P\left(A|B\right)=\frac{2}{3}$
4. $P\left(A^{\prime }|B^{\prime }\right)=\frac{1}{3}$

Q. If A and B are independent events such that
$P\left(A\right)=p,P\left(B\right)=2p$ and
$P\left(\text{Exactly one of A and B}\right)=\frac{5}{9}$
then $p=$

1. $\frac{1}{3}$
2. $\frac{5}{6}$
3. $\frac{5}{12}$
4. $\frac{9}{11}$

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

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

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