Conditional Probability

Q.

From a class of $12$ girls and $18$ boys, two students are chosen randomly. What is the probability that both of them are girls?

1. $\frac{22}{145}$

2. $\frac{13}{15}$

3. $\frac{1}{18}$

4. None of these.

Q.

A biased coin with probability $p,0<p<1$ of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is$2/5$, then $p$ equals

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

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

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

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

Q. Let $A$ and $B$ be two non-null events such that $A\subset B.$ Then, which of the following statements is always correct?

1. $P\left(A|B\right)=1$
2. $P\left(A|B\right)\le P\left(A\right)$
3. $P\left(A|B\right)\ge P\left(A\right)$
4. $P\left(A|B\right)=P\left(B\right)-P\left(A\right)$

Q.

A contest consists of predicting the results (win, draw or defeat) of $10$ football matches. The probability that one entry contains at least $5$ correct answers is

1. $\frac{12585}{3^{10}}$

2. $\frac{12385}{3^{10}}$

3. $\frac{9385}{3^{10}}$

4. None of these

Q. A randomly selected year is containing 53 Mondays then probability that it is a leap year

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

Q. In a college, 30% students fail in physics, 40% fail in mathematics and 20% fail in both. One student is chosen at random. The probability that she fails in mathematics if she has failed in physics is

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

Q. Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting equal numbers on the dice is

1. 0.2
2. 0.1
3. 0.5
4. 0.25

Q. Let $n_1$ and $n_2$ be the number of red and black balls, respectively in box I. Let $n_3$ and $n_4$ be the number of red and black balls, respectively in box II.
A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$, then the correct option(s) with the possible values of $n_1$ and $n_2$ is/are

1. $n_1=4and{n}_2=6$
2. $n_1=2and{n}_2=3$
3. $n_1=10and{n}_2=20$
4. $n_1=3and{n}_2=6$

Q. A hunter's chance of shooting an animal at a distance $r$ is $\frac{a^2}{r^2}(r>a).$ He fires when $r=2a$ if he misses he reloads fires when $r=3a,4a,\cdots$. If he misses at a distance $na$, the animal escapes, then the odds against the hunter is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $\frac{n+1}{n-1}$
2. $\frac{n-1}{n+1}$
3. $\frac{n-1}{n}$
4. $\frac{n+1}{n}$

Q. Two fair dice are rolled simultaneously. It is found that one of the dice show odd prime number. The probability that the remaining dice also show an odd prime number, is equal to

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

Q. Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

1. $\frac{1}{10}$
2. $\frac{1}{11}$
3. $\frac{1}{12}$
4. $\frac{1}{17}$

Q. Two numbers are randomly selected and multiplied. Consider two events $E_1$ and $E_2$ defined as
$E_1$ : Their product is divisible by 5
$E_2$ : Unit’s places in their product is 5
Which of the following statement is/are correct?

1. $E_1$ is twice as likely to occur as $E_2$
2. $E_1$ and $E_2$ are disjoint
3. $P\left(E_2/E_1\right)=\frac{1}{4}$
4. $P\left(E_1/E_2\right)=1$

Q. A number is randomly selected from the first 40 natural numbers. What is the probability that the selected number is divisible by 5 or 7?

1. $\frac{8}{20}$
2. $\frac{17}{40}$
3. $\frac{7}{20}$
4. $\frac{3}{10}$

Q. A number is randomly selected from the first 40 natural numbers. What is the probability that the selected number is divisible by 5 or 7?

1. $\frac{8}{20}$
2. $\frac{17}{40}$
3. $\frac{7}{20}$
4. $\frac{3}{10}$

Q. Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting equal numbers on the dice is

1. 0.2
2. 0.1
3. 0.5
4. 0.25

Q. Let $A$ and $B$ be two events such that $P\left(A\right)=\frac{3}{8},P\left(B\right)=\frac{1}{2}$ and $P(A\cup B)=\frac{5}{8}$. Then which of the following do/does hold good?

1. $P\left(A^C|B\right)=2P\left(A|B^C\right)$
2. $P\left(B\right)=P\left(A|B\right)$
3. $15P\left(A^C|B^C\right)=8P\left(B|A^C\right)$
4. $P\left(A|B^C\right)=P(A\cap B)$

Q. Two numbers are randomly selected and multiplied. Consider two events $E_1$ and $E_2$ defined as
$E_1$ : Their product is divisible by 5
$E_2$ : Unit’s places in their product is 5
Which of the following statement is/are correct?

1. $E_1$ is twice as likely to occur as $E_2$
2. $E_1$ and $E_2$ are disjoint
3. $P\left(E_2/E_1\right)=\frac{1}{4}$
4. $P\left(E_1/E_2\right)=1$

Q. Two integers are selected at random from the set $\{1,2,\dots ,11\}.$ Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :

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

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. Let $A$ and $B$ be two non-null events such that $A\subset B.$ Then, which of the following statements is always correct?

1. $P\left(A|B\right)=1$
2. $P\left(A|B\right)\le P\left(A\right)$
3. $P\left(A|B\right)\ge P\left(A\right)$
4. $P\left(A|B\right)=P\left(B\right)-P\left(A\right)$

Q. Three numbers are choosen at random without replacement from {1, 2, 3, ....8}. The probability that their minimum is 3, given that their maximum is 6, is

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

Q. $a>b,k>0\Rightarrow \{\begin{array}{c}ak>bk\\ -ak<-bk\end{array}$

1. True
2. False

Q. A and B are events such that $P\left(A\right)=0.3,P(A\cup B)=0.8$. If A and B are independent then P(B)=

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

Q. The probabilities of different faces of a biased dice to appear are as follows
$\begin{array}{ccccccc}Facenumber& 1& 2& 3& 4& 5& 6\\ Probability& 0.1& 0.32& 0.21& 0.15& 0.05& 0.17\end{array}$
The dice is thrown and it is known that either the face number 1 or 2 will appear. Then, the probability of the face number 1 to appear is

1. $\frac{5}{21}$
2. $\frac{5}{13}$
3. $\frac{7}{23}$
4. $\frac{3}{10}$

Q.

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

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

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

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

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

Q. One ticket is selected at random from $100$ tickets numbered $00,01,02,\cdots ,98,99$. If $x_1$ and $x_2$ denotes the sum and product of the digits on the tickets, then $P(x_1=9/x_2=0)$ is equal to

1. $\frac{2}{19}$
2. $\frac{19}{100}$
3. $\frac{1}{50}$
4. None of these

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. Two dice are thrown. The probability that the sum of numbers coming up on them is 9, if it is known that the number 5 always occurs on the first die, is

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

Q.

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

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

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

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

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