Probability Questions (959)

For three independent events A, B and C, the probability of exactly one of the events A or B occurring = the probability of exactly one of the events B or C occurring = the probability of exactly one of the events C or A occurring = p. If the probability of all the events occurring simultaneously be p2 where 0 < p < 0.5, then find the probability of at least one of the events A, B and C occurring.

Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a "best of 3 games" or a "best of 5 games" match against B, which option should be chosen so that the probability of his winning the match is higher? (No game ends in a draw.)

Urn A contains 6 red and 4 black balls and urn B contains 4 red and 6 black balls. One ball is drawn at random from urn A and placed in urn B. Then one ball is drawn at random from urn B and placed in urn A. If one ball is now drawn at random from urn A, the probability that it is red is:

If three squares are selected at random from chessboard, then the probability that they form the letter "L" is

Let AB be a line segment of length a divided at P and Q such that AP = x and BQ = y. A point is selected at random inside the triangle formed by the conditions. What is the probability that x < a/2, y < a/2, PQ < a/2? (Given answer: 0.25)

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is

For Problems 1–3: In a class of 10 students, probability of exactly i students passing an examination is directly proportional to i2. Then answer the following questions:The probability that exactly 5 students passing an examination is

Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a dice. Find the probability of(i) the dice shows a number greater than 4(ii) there is at least one tail

If \(A\) and \(B\) are two independent events such that \(P(\bar{A} \cap B) = 2/15\) and \(P(A \cap \bar{B}) = 1/6\), then \(P(B)\) is

In a binomial distribution \(B(b,\, p = 1/4)\), if the probability of at least one success is greater than or equal to 9/10, then \(n\) is greater than

A natural number is selected at random from the set \(X = \{x \mid 1 \leq x \leq 100\}\). The probability that the number satisfies the inequation \(x^2 - 13x \leq 30\), is

If the mean and the variance of a binomial variate \(X\) are 2 and 1, respectively, then the probability that \(X\) takes a value greater than or equal to one is

Two numbers are chosen from {1, 2, 3, 4, 5, 6, 7, 8} one after another without replacement. Then the probability that(1) the smaller value of two is less than 3 is 13/28(2) the bigger value of two is more than 5 is 9/14(3) product of two number is even is 11/14(4) none of these

A random variable \(X\) has Poisson distribution with mean 2. \(P(X > 1.5)\) equals

A coin is tossed three times. Let \(A\): at most two tails, \(B\): at least one tail. Find \(P(A/B)\).

Entries of a $2 \times 2$ determinant are chosen from the set $\{-1, 1\}$. The probability that determinant has zero value is:

\(A\) and \(B\) are two independent events. The probability that both \(A\) and \(B\) occur is \(\frac{1}{6}\) and the probability that neither of them occur is \(\frac{1}{3}\). Then \(P(A)\) is equal to

A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and tested till all the defective articles are found. What is the probability that the testing procedure ends at the 12th testing?

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

$A$ and $B$ play a game of tennis. The situation of the game is as follows; if one scores two consecutive points after a deuce he wins; if loss of a point is followed by win of a point, it is deuce. The chance of a server to win a point is $2/3$. The game is at deuce and $A$ is serving. Probability that $A$ will win the match is : (Serves are changed after each game)

If the probability of a six-digit number \(N\) whose six digits are 1, 2, 3, 4, 5, 6 written as random order is divisible by 6 is \(p\), then the value of \(1/p\) is ________.

A business man is expecting two telephone calls. Mr Walia may call any time between 2 p.m and 4 p.m. while Mr Sharma is equally likely to call any time between 2.30 p.m. and 3.15 p.m. The probability that Mr Walia calls before Mr Sharma is :

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equal to :

From $4m+1$ tickets numbered as $1, 2, \ldots 4m+1$. Three tickets are chosen at random. The probability that the numbers are in A.P. with even common difference is

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

Four die are thrown simultaneously. The probability that 4 and 3 appear on two of the die given that 5 and 6 have appeared on other two die is

Three randomly chosen non-negative integers \(x\), \(y\) and \(z\) are found to satisfy the equation \(x + y + z = 10\). Then the probability that \(z\) is even, is

For Problems 10–12: Let \(S\) and \(T\) are two events defined on a sample space with probabilities \(P(S) = 0.5\), \(P(T) = 0.69\), \(P(S/T) = 0.5\).Events \(S\) and \(T\) are

Four persons can hit a target correctly with probabilities \(1/2, 1/3, 1/4\) and \(1/8\) respectively. If all hit at the target independently, then the probability that the target would be hit, is:

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:

In a class of 125 students 70 passed in Mathematics, 55 in Statistics, and 30 in both. Then find the probability that a student selected at random from the class has passed in only one subject.

A binary number is made up of 8 digits. Suppose that the probability of an incorrect digit appearing is \(p\) and that the errors in different digits are independent of each other. Then find the probability of forming an incorrect number.

$2^n$ players of equal strength are playing a knock out tournament. If they are paired randomly in all rounds, the probability that out of two particular players $S_1$ and $S_2$ exactly one will reach in semi final is $(n \in N, n \geq 2)$:

Consider the following two statements:Statement-1: For events \(A\) and \(E\),\[P(E/A) \geq P(A/E) \cdot P(E)\]Statement-2: \(P(A/E) \geq P(A \cap E)\)Which of the following is correct?

State whether the statements are true or false.The probabilities of the events \(A\), \(B\) and \(C\) are respectively \(\frac{3}{2}\), \(\frac{1}{4}\) and \(\frac{1}{6}\). Then the events are mutually exclusive.

Three identical dice are rolled. The probability that the same number will appear on each of them is

Two integer $'a'$ and $'b'$ are randomly selected from the set $\{1, 2, \ldots\}$ (with replacement) then if the probability of $\frac{1}{5}(a^2 + b^2)$ being positive integer is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - 2p = \ldots\ldots\ldots\ldots$

If \(p\) and \(q\) are chosen randomly from the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) with replacement, then determine the probability that the roots of the equation \(x^2 + px + q = 0\) are real.

A bag contained 3 maths book and 2 physics books. A book is drawn at random if it is of math, 2 more books of maths together with this book put back in the bag and if it is of physics it is not replaced in the bag. This experiment is repeated 3 time. If third draw gives math book, The probability that first two drawn books were of physics is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - 8(p+1) = \ldots\ldots\ldots\ldots$

Two friends decide to meet at a spot between 2 p.m. and 3 p.m. whosoever arrives first agrees to wait for 15 minutes for the other. The probability that they met is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - p = \ldots\ldots\ldots\ldots$

There are two purses. The first contains 9 fifty paise coins and a one-rupee coin, while the second purse has 10 fifty-paise coins. Nine coins are transferred from the first purse to the second randomly. Then nine coins are transferred from the second purse to the first randomly. The probability of finding a one rupee coin in the first purse after these transfers is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - p = \ldots\ldots\ldots\ldots$

An ordinary coin is tossed n times. If the probability of getting 7 heads is equal to that of 9 heads then the probability of getting 2 heads is

A dice is loaded so that the probability of a face i is proportional to i², i = 1, 2, …, 6. Then find the probability of occurring a prime number when the dice is rolled.

Raj and Sanchita are playing game in which they throw a die alternately till one of them gets a six. The probability that Sanchita win the game is $p$ then the value of $5p?$

Rajesh doesn't like to study. Probability that he will study is $1/3$. If he studied, then probability that he will fail is $1/2$ and if he didn't study then probability that he will fail is $3/4$. If in result Rajesh failed, then what is the probability that he didn't study.

In a board exam there are two sections each section has $5$ questions. As per the given condition a candidate has to answer any $6$ questions out of $10$ questions. What is the probability that a student answered $6$ questions such that not more than $4$ questions selected from one section?

On a rod of length $6$ units, lengths $1, 2$ units are measured at random, the probability that no points of the measured lines will coincide is ______.

Assume that the birth of a boy or girl to a couple to be equally likely, and exhaustive. For a couple having $6$ children, the probability that their "three oldest are boys" is $p$ then the value of $10p$.

The probability of randomly drawing five cards from a deck that has exactly one Ace is ______.

In a poker game, the probability of getting a "pair" in five cards is ______. [A pair consists of $2$ cards of the same kind (eg. $2$ kings) and $3$ cards that are different from the kind of the pair (eg. different from kings) and that are all different from each other.]