Probability of an event; addition and multiplication theorems — practice questions

Question 1 · difficulty L2 · understanding

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $$i^{\text {th }}$$ roll than the number obtained in the $$(i-1)^{\text {th }}$$ roll, $$i=2,3$$, is equal to

• A. 5/54
• B. 2/54
• C. 1/54
• D. 3/54

Question 2 · difficulty L2 · understanding

Two integers $$x$$ and $$y$$ are chosen with replacement from the set $$\{0,1,2,3, \ldots, 10\}$$. Then the probability that $$|x-y|>5$$, is :

• A. $$\frac{31}{121}$$
• B. $$\frac{60}{121}$$
• C. $$\frac{62}{121}$$
• D. $$\frac{30}{121}$$

Question 3 · difficulty L2 · understanding

Three dice are rolled. If the probability of getting different numbers on the three dice is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q-p$$ is equal to :

• A. 3
• B. 4
• C. 1
• D. 2

Question 4 · difficulty L2 · understanding

Two dies 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 :

• A. $${4 \over 9}$$
• B. $${1 \over 2}$$
• C. $${5 \over {12}}$$
• D. $${17 \over {36}}$$

Question 5 · difficulty L2 · understanding

A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is ______________.

Question 6 · difficulty L2 · understanding

A number of chosen at random from the set {1, 2, 3, ....., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is __________.

Question 7 · difficulty L3 · understanding

Two distinct numbers $a$ and $b$ are selected at random from $1,2,3, \ldots, 50$. The probability, that their product $a b$ is divisible by 3 , is

• A. $\frac{272}{1225}$
• B. $\frac{561}{1225}$
• C. $\frac{664}{1225}$
• D. $\frac{8}{25}$

Question 8 · difficulty L3 · understanding

Two number $\mathrm{k}_1$ and $\mathrm{k}_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $\mathrm{i}^{\mathrm{k}_1}+\mathrm{i}^{\mathrm{k}_2},(\mathrm{i}=\sqrt{-1})$ is non-zero, equals

• A. $\frac{3}{4}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $\frac{2}{3}$

Question 9 · difficulty L3 · understanding

Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a square matrix of order 2 with entries either 0 or 1 . Let E be the event that A is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is :

• A. $\frac{3}{8}$
• B. $\frac{1}{8}$
• C. $\frac{3}{16}$
• D. $\frac{5}{8}$

Question 10 · difficulty L3 · understanding

$A$ and $B$ alternately throw a pair of dice. A wins if he throws a sum of 5 before $B$ throws a sum of 8 , and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5 . The probability, that A wins if A makes the first throw, is

• A. $\frac{8}{19}$
• B. $\frac{9}{19}$
• C. $\frac{8}{17}$
• D. $\frac{9}{17}$