Conditional Probability

Q. In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together ?

ANS : 14400
PLEASE ELABORATE THE LOGICAL CONSTRUCTION TO REACH ANSWER

Q.

Four boys and three girls stand in a queue for an interview, the probability that they will be in alternate positions is

1. $\frac{1}{34}$

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

3. $\frac{1}{17}$

4. $\frac{1}{68}$

Q.

The length, breadth, and height of a room are $825\mathrm{cm},675\mathrm{cm}\mathrm{and}450\mathrm{cm}$ respectively. The longest tape which can measure the three dimensions of the room exactly.

Q.

If $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$, and $P(A\cap B)=\frac{7}{12}$, then the value of $P\left(A^{}\cap B^{}\right)$ is

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

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

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

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

Q.

A pair has two children. If one of them is boy, then the probability that other is also a boy is

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

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

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

4. None of these.

Q.

8 coins are tossed simultaneously. The probability of getting at least 6 heads is:

1. 

2. 

3. 

4. 

Q.

The probability of a man hitting a target is $\frac{1}{10}$. The least number of shots required, so that the probability of his hitting the target at least once is greater than $\frac{1}{4}$ is

1. $4$

2. $3$

3. $2$

4. $1$

Q. A box contains three coins: two regular coins and one fake two-headed coin (P(H)=1)
You pick a coin at random and toss it, and get heads. The probability that it is the two-headed coin = ___

Q.

Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday).

Each is equally likely to visit the shop on any day as on another day.

What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?

Q.

The probability that the three cards drawn from a pack of $52$ cards are all red is

1. $\frac{1}{17}$

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

3. $\frac{2}{19}$

4. $\frac{2}{17}$

Q.

A standard deck of cards consists of four suits (clubs, diamonds, hearts, and spades), with each suit containing $13$ cards (ace, two through ten, jack, queen, and king) for a total of $52$ cards in all. How many $7$ card hands will consist of exactly $2$ hearts and $3$ clubs?

Q. How many even numbers of 3 different digits can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition is not allowed)

1. 224
2. 280
3. None of these
4. 324

Q.

If a dice is thrown $7$ times, then the probability of obtaining $5$ exactly $4$ times is

1. ${}^{7}C_4{\left(\frac{1}{6}\right)}^4{\left(\frac{5}{6}\right)}^3$

2. ${}^{7}C_4{\left(\frac{1}{6}\right)}^3{\left(\frac{5}{6}\right)}^4$

3. ${\left(\frac{1}{6}\right)}^4{\left(\frac{5}{6}\right)}^3$

4. ${\left(\frac{1}{6}\right)}^3{\left(\frac{5}{6}\right)}^4$

Q.

In a book of $500$ pages, it is found that there are $250$ typing errors.

Assume that Poisson law holds for the number of errors per page.

Then, the probability that a random sample of $2$ pages will contain no error, is

1. $e^{-0.3}$

2. $e^{-0.5}$

3. $e^{-1}$

4. $e^{-2}$

Q.

A problem in statistics is given to five students A, B, C, and D. Their chances of solving it are $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$ and $\frac{1}{6}$ respectively. What is the probability that the problem will be solved?

Q.

A student answers a multiple choice question with $5$ alternatives, of which exactly one is correct. The probability that he knows the correct answer is $p$, $0<p<1$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly, is

1. $\frac{3p}{4p+3}$

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

3. $\frac{5p}{4p+1}$

4. $\frac{4p}{3p+1}$

Q.

Two dice are rolled.

If both dices have six faces numbered $1,2,3,5,7$ and $11$, then the probability that the sum of the numbers on the top faces is less than or equal to $8$ is

1. $\frac{17}{36}$

2. $\frac{4}{9}$

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

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

Q.

Out of 10 students there are 6 girls and 4 boys.A team of 4 students is selected at random.Find the probability that there are at least 2 girls.

Q.

A bag contains 3 red, 7 white and 4 black balls. If three balls are drawn from

the bag, then the probability that all of them are of the same colour is

1. 

2. 

3. 

4. None of these

Q.

Out of $30$ consecutive numbers, $2$ are chosen at random. The probability that their sum is odd is

1. $\frac{14}{29}$

2. $\frac{16}{29}$

3. $\frac{15}{29}$

4. $\frac{10}{29}$

Q. If two events A and B are such that $P\left(A^C\right)=0.3,P\left(B\right)=0.4,P(A\cap B^C)=0.5$, then find the value of $P\left[\frac{B}{A\cup B^C}\right]$

1. 
2. 
3. 
4. 

Q. A die is thrown three times. Events $A$ and $B$ are defined as below:
$A:4$ on the third throw
$B:6$ on the first and $5$ on the second throw.
The probability of occurrence of $A$ given that $B$ has already occurred is:

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

Q.

It is given that the events $A$and $B$ are such that$P\left(A\right)=\frac{1}{4}$, $P\left(\frac{A}{B}\right)=\frac{1}{2}$ and$P\left(\frac{B}{A}\right)=\frac{2}{3}$. Then, $P\left(B\right)$ is equal to

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

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

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

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

Q.

There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows head, what is the probability that it is was the two headed coin?

Q. If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find (i) P(A ∩ B) (ii) P(A|B) (iii) P(A ∪ B)

Q.

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Q. A bag contains $5$ red marbles and $3$ black marbles. Three marbles are drawn one by one without replacement. What is the probability that atleast one of the three marbles drawn be black, if the first marble is red?

Q. Let $S$ be the sample space of all $3\times 3$ matrices with entries from the set {0, 1}. Let the events $E_1$ and $E_2$ be given by
$E_1=\{A\in S:detA=0\}$ and
$E_2=${A $\in S:$ sum of entries of $A$ is $7$}
If the matrix is chosen at random from $S,$ then the conditional probability $P\left(E_1|E_2\right)$ equals

Q. A speaks truth in 75% of the cases, while B
in 90% of the cases.In what percent of cases are they likely to contradict each other in stating the same fact?

Q. A black and a red dice are rolled. (a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. (b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.