Probability Practice

1) Out of $21$ tickets consecutively numbered , three are drawn at random . Find the probability that the numbers on them are in A.P.

2) If $2$ points are selected on a line of length L so as to be on opposite sides of the midpoint of the line , find the probability that the distance between the $2$ points is greater than $\dfrac{L}{3}$ .

3) If n points are independently chosen at random on the circumference of a circle , what is the probability that these points lie in some semicircle.

4) A coin is tossed $m + n$ times $(m >n)$ . What is the probability that at least m consecutive heads come up.

5) From an urn containing six balls, $3$ white and $3$ black ones, a p erson selects at random an even number of balls (all the different ways of drawing an even number of balls are considered equally probable, irrespective of their number). Then the probability that there will be the same number of black and white balls among them.

6) $5$ different marbles are placed in $5$ different boxes randomly. Find the probability that exactly two boxes remain empty. (each box can hold any number of marbles)

7) Two red counters, three green counters and $4$ blue counters are placed in a row in random order. The probability that no two blue counters are adjacent is

8) Find the probability of ruin of each of $2$ players when they continue a certain game till the ultimate ruin of on of them.The winner of each game gets one ruble.

9) Let $A$ and $B$ be $2$ independent witnesses in a case . The probability that $A$ will speak the truth is x an the probability that $B$ will speak the truth is y. $A$ and $B$ agree in a certain statement. Probability of true statement-

10) A artillery target may be either at a point $A$ with probability $\dfrac{8}{9}$ or at a point $B$ with probability $\dfrac{1}{9}$ . We have 21 shells each of which can be fixed either at point $A$ or $B$ . Each shell may hit the target independently of the other with probability $\dfrac{1}{2}$ . How many shells must be fixed at point A to hit the target with maximum probability.

11) If $a$ and $b$ are chosen randomly from the set consisting of numbers $1 , 2, 3, 4, 5, 6$ with replacement . Find the probability that $\displaystyle lim_{x \to 0} \left( \dfrac{a^x + b^x}{2} \right)^{\dfrac{2}{x}} = 6$

12) From the set $S = ( 1 , 2 , 3 , \cdots , 3n )$ , three numbers are chosen at random . Find the probability that the sum of the chosen numbers is divisible by 3.

13) Two players $P_{1}$ and $P_{2}$ are playing the fina of a chess championship , which consisits of a series of matches . Probability of $P_{1}$ winning a match is $\dfrac{2}{3}$ and for $P_{2}$ is $\dfrac{1}{3}$ . The winner will be the one who is ahead by 2 games as compared to the other player and wins atleast $6$ games . Now , if the player $P_{2}$ wins first four matches , find the probability of $P_{1}$ winning the championship.

14) Let $X$ be a set containing n elements . Find the number of all ordered triplets $(A, B , C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$ .

15) A point is selected at random from the interior of the pentagon with vertices $A = (0,2) , B = (4 , 0) , C = (2 \pi + 1 , 0) , D = ( 2 \pi + 1 , 4) , E = (0 , 4)$ . What is the probability that angle $APB$ is obtuse .

16) There are two bags , each containing $5$ red and $3$ black balls . Two persons $A$ and $B$ are given one bag each . Each of them is to draw one ball at random from the bag till both of them get a black ball ( not necessary in the same draw). The balls are to be replaced after each draw . Find the probability that the number of trials required is $n.$

17) Each of n urns contains $a$ white and $b$ blac balls . One ball is transferred from the first urn into the second , then one ball from the latter into the third and so on . If finally one ball is taken out from the last urn , then what is the probability of its being black.

18) In a knockout tournament , $2^n$ equally skilled players namely $S_{1} , S_{2} , S_{3} , \cdots , S_{2^n}$ are participating. In each round , players are divided in pairs at random and winner from each pair moves in the next round . If $S_{2}$ reaches semi - final , then find the probability that $S_{1}$ will win the tournament.

19) If a pair of dice is thrown twice , then what is the probability that the sum of outcomes of each of the two throws are equal.

20) Two teams $A$ and $B$ play tournament. The first one to win $(n+1)$ games , win the series . The probability that $A$ wins a game is $p$ and that $B$ wins a game is $q$ ( no ties) . Find the probability that A wins the series.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Probability Practice

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