Conditional Probability

Q.

A contest consists of predicting the results of the win, draw or defeat of $7$ football matches. A sent his entry by predicting at random. The probability that his entry will contain exactly $4$ correct predictions is

1. $\frac{8}{3^7}$

2. $\frac{16}{3^7}$

3. $\frac{280}{3^7}$

4. $\frac{560}{3^7}$

Q.

A bag $X$ contains $2$ white and $3$ black balls and another bag $Y$ contains $4$ white and $2$ black balls. One bag is selected at random and a ball is drawn from it. Then the probability for the ball chosen be white is

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

2. $\frac{7}{15}$

3. $\frac{8}{15}$

4. $\frac{14}{15}$

Q. A box contains 5 green, 4 yellow and 3 white marbles. Three marbles are drawn at random. What is the probability that they are not of the same colour ?

1. $\frac{41}{44}$
2. $\frac{40}{39}$
3. $\frac{40}{44}$
4. $\frac{44}{41}$
5. $\frac{44}{39}$

Q.

If $\text{A}$ and $\text{B}$ are two events such that$\text{P(AUB)}$$+$$\text{P(A∩B)}$$=$$\frac{7}{8}$, and $\text{P(A)}$$=$$\text{2P(B)}$, then $\text{P(A)}$$=$

1. $\frac{7}{12}$

2. $\frac{7}{24}$

3. $\frac{5}{12}$

4. $\frac{17}{24}$

Q. A box contains 6 blue, 5 green and 4 red balls. If two balls are picked at random, then what is the probability that neither is blue?

1. None of these
2. $\frac{12}{35}$
3. $\frac{11}{21}$
4. $\frac{10}{21}$
5. $\frac{3}{5}$

Q. A Number is selected at random from first thirty natural numbers. What is the chance that it is a multiple of either 3 or 13?

1. 
2. 
3. 
4. 

Q.

A box contains $100$ bulbs, out of which $10$ are defective. A sample of $5$ bulbs is drawn. The probability that none is defective, is

1. $\frac{9}{10}$

2. ${\left(\frac{1}{10}\right)}^5$

3. ${\left(\frac{9}{10}\right)}^5$

4. ${\left(\frac{1}{2}\right)}^5$

Q.

A bag contains $n$ white and $n$ red balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each pair consists of one white and one red ball is

1. $\frac{2n}{{}^{2n}C_n}$

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

3. $\frac{1}{{}^{2n}C_n}$

4. None of these

Q.

Ten students are seated at random in a row. The probability that two particular students are not seated side by side is

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

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

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

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

Q.

A natural number is chosen at random from amongst the first $300$. What is the probability that the number chosen is a multiple of $2\mathrm{or}3\mathrm{or}5$ ?

1. $\frac{1}{10}$

2. $\frac{11}{15}$

3. $\frac{4}{150}$

4. $\frac{17}{30}$

Q. A box contains 2 blue, 3 green and 5 red balls. If three balls are drawn at random, what is the probability that all balls are different in color?

1. $\frac{1}{4}$
2. $\frac{3}{10}$
3. $\frac{3}{7}$
4. $\frac{4}{11}$
5. $\frac{2}{9}$

Q.

A letter is taken out at random from ‘$ASSISTANT$’ and another is taken out from ‘$STATISTICS$’. The probability that they are the same letters, is

1. $\frac{1}{45}$

2. $\frac{13}{90}$

3. $\frac{19}{90}$

4. None of these

Q. There are 8 brown balls, 4 oragne balls and 5 black balls in a bag. Five balls are chosen at random. What is the probability of there being 2 brown balls, 1 orange ball and 2 black balls?

1. $\frac{191}{1547}$
2. $\frac{180}{1547}$
3. $\frac{280}{1547}$
4. $\frac{180}{1547}$
5. None of these

Q.

A bag contains 3 white, 5 red and 7 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is:

1. 3/22

2. 1/91

3. none

4. 1/22

5. 2/91

Q.

For$k=1,2,3$ the box, $Bk$ contains $k$ red balls and $(k+1)$white balls. Let $P\left(B_1\right)=\frac{1}{2},P\left(B_2\right)=\frac{1}{3}$and $P\left(B_3\right)=\frac{1}{6}$.

A box is selected at random and a ball is drawn from it.

If a red ball is drawn, then the probability that it has come from box $B_2$ is

1. $\frac{35}{78}$

2. $\frac{14}{39}$

3. $\frac{10}{13}$

4. $\frac{12}{13}$

Q. If 3 balls are drawn at random, what is the probability that 1 is green and the other 2 are black ?

1. $\frac{4}{87}$
2. $\frac{2}{85}$
3. $\frac{1}{85}$
4. $\frac{3}{85}$
5. $\frac{5}{85}$

Q. If 5 balls are picked at random, what is the probability that none are orange ?

1. $\frac{33}{442}$
2. $\frac{442}{33}$
3. $\frac{15}{167}$
4. $\frac{23}{235}$
5. $\frac{35}{235}$

Q.

Two dice are tossed simultaneously. Then the probability that the total score is a prime number is:

1. none

2. 5/12

3. 1/2

4. 1/12

5. 7/12

Q.

The probability that in a random arrangement of the letters of the word 'ANIMAL', the two A's come together.

1. none

2. 4/5

3. 2/5

4. 2/3

5. 1/3

Q.

A number is selected randomly from the first thirty natural numbers. What are the chances that it is a multiple of either 3 or 13?

1. 13/30

2. 2/5

3. 3/5

4. 17/30

5. None of these

Q.

x${}_1$, x${}_2$, x${}_3$, ..., x${}_{40}$ are forty real numbers such that x${}_r$$<$x${}_{r+1}$for r = 1, 2, 3... 39. Five numbers out of these are picked up at random. The probability that the five numbers have x${}_{20}$ as the middle number is:

1. 19/55

2. (¹⁹C₂ײ⁰C₂)/( ⁴⁰C₄)

3. (¹⁹C₂ײ⁰C₂)/( ⁴⁰C₅)

4. (¹⁹C₃ײ⁰C₂)/( ⁴⁰C₄)

5. None of these

Q. What is the probability that product of two integers chosen at random has the same unit’s digit as the integers themselves?

1. 
2. 
3. 
4. 

Q.

An urn contains 6 red, 4 blue and 3 green marbles. If 2 marbles are picked at random, then what is the probability that both the marbles are red?

1. 

2. 

3. 

4. 

Q. If $A$ and $B$ are two events such that $P(A\cup B)=\frac{5}{6},P(A\cap B)=\frac{1}{3}$ and $P\left(\overline{B}\right)=\frac{1}{3}$, then the value of $P\left(A\right)$ is

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

Q. Arun draws 3 balls at random from a basket which contains 4 red and 5 blue balls. What are the odds in favour of these being all blue?

1. 5 : 42
2. 5 : 37
3. 4 : 43
4. 4 : 50

Q. $10$ different books and $2$ different pens are given to $3$ boys so that each gets equal number of things. The probability that the same boy does not receive both the pens is

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