Probability Questions (959)

A man and a woman appear in an interview for two vacancies in the same post. The probability of man's selection is 1/4 and that of the woman's selection is 1/3. What is the probability that none of them will be selected?

A hat contains a number of cards with 30% white on both sides, 50% black on one side and white on the other side, 20% black on both sides. The cards are mixed up, and a single card is drawn at random and placed on the table. Its upper side shows up black. The probability that its other side is also black is

Of the three independent events \(E_1\), \(E_2\), and \(E_3\), the probability that only \(E_1\) occurs is \(\alpha\), only \(E_2\) occurs is \(\beta\), and only \(E_3\) occurs is \(\gamma\). Let the probability \(p\) that none of the events \(E_1\), \(E_2\), or \(E_3\) occurs satisfy the equations \((\alpha - 2\beta)\,p = \alpha\beta\) and \((\beta - 3\gamma)\,p = 2\beta\gamma\). All the given probabilities are assumed to lie in the interval \((0, 1)\). Then \[\frac{\text{Probability of occurrence of } E_1}{\text{Probability of occurrence of } E_3} = \underline{\hspace{2cm}}.\] (JEE Advanced 2013)

Two fair dice are thrown till outcome is 12. What is the probability that one has to do 20 throws for this?

In a bag there are six balls of unknown colours, three balls are drawn at random and found to be all black. Find the probability that no black ball is left in the bag. (Or, Find the probability that the bag contained exactly 3 black balls.)

A person goes to office either by car, scooter, bus or train, the probability of which being 1/7, 3/7, 2/7 and 1/7, respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is 2/9, 1/9, 4/9 and 1/9 respectively. Given that he reached office in time, what is the probability that he traveled by a car?

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is __________.

If \(p\) is the probability that a man aged \(x\) will die in a year, then the probability that out of \(n\) men \(A_1, A_2, \ldots, A_n\) each aged \(x\), \(A_1\) will die in an year and be the first to die is

A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is

Let \(E\) and \(F\) be two independent events. The probability that both \(E\) and \(F\) happen is \(1/12\) and the probability that neither \(E\) nor \(F\) happens is \(1/2\), then a value of \(P(E)/P(F)\) is

What is the probability of guessing correctly at least 8 out of 10 answers on a true–false examination?

Events A and B are such that \(P(A) = 1/2\), \(P(B) = 7/12\) and \(P(\text{not } A \text{ or not } B) = 1/4\). State whether A and B are independent?

If the papers of 4 students can be checked by any one of the 7 teachers, then the probability that all the 4 papers are checked by exactly 2 teachers is

A bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw till one of them draws a white ball and wins the game. A begins the game. If the probability of A winning the game is three times that of B, then find the ratio \(a : b\).

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2$, $\sqrt{3N}$, $N+2$ are in geometric progression be $\frac{k}{48}$. Then the value of k is

Two dices are rolled one after the other. The probability that the number on the first is smaller than the number on the second is

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:

Three persons A, B and C, in order, cut a pack of cards replacing them after each cut on the condition that the first who cuts a spade shall win the prize. Find their respective chances.

A man takes a step forward with probability 0.4 and backward with probability 0.6. Then find the probability that at the end of eleven steps he is one step away from the starting point.

A number x is chosen at random from the set \(\{1, 2, 3, 4, \ldots, 100\}\). Define the event: \(A\) = the chosen number x satisfies \(\dfrac{(x-10)(x-50)}{(x-30)} \geq 0\). Then \(P(A)\) is

In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of 3 girls sitting together in a back row on adjacent seats?

The probability so that $r$ $1 \times 1$ squares which are selected from a $m \times n$ chess board such that no two of them share the same row or same column is :

Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up, respectively, is

For Problems 19–21: A player tosses a coin and scores one point for every head and two points for every tail that turns up. He plays on until his score reaches or passes \(n\). \(P_n\) denotes the probability of getting a score of exactly \(n\).The value of \(P_n\) is equal to

In an organization number of women are $\mu$ times that of men. If $\alpha$ things are to be distributed among them than the probability that the number of things received by men are odd is $\left(\frac{1}{2} - \left(\frac{1}{2}\right)^{\alpha+1}\right)$. Then $\mu = \ldots\ldots\ldots\ldots$

A pack of playing cards was found to contain only 51 cards. If the first 13 cards, which are examined, are all red, what is the probability that the missing card is black?

If \(a\) and \(b\) are chosen randomly by throwing a pair of fair dice, then the probability that \(\lim_{x\to 0}\left(\dfrac{a^x + b^x}{2}\right)^{\frac{2}{x}} = 6\) equals:

A = even that the item came from lot \(A\); \(P(A) = \dfrac{3}{7}\). \(B\) = item came from \(B\); \(P(B) = \dfrac{4}{7}\). \(D\) = item from mixed lot 'C' is defective. \(P(D) = P(D \cap A) + P(D \cap B) = P(A) \cdot P(D/A) + P(B) \cdot P(D/A)\). Find \(P(D)\).

In a match, the probability that team W wins is \(\frac{1}{2}\) and the probability that team L wins is \(\frac{1}{2}\). Two matches are played. Find the probability that the results are \(W_1 L_2\) or \(L_1 W_2\) (i.e., the teams split the wins).

Die \(A\) has 4 red and 2 white faces, whereas die \(B\) has 2 red and 4 white faces. A coin is flipped once. If it shows a head, the game continues by throwing die \(A\); if it shows tail, then die \(B\) is to be used. If the probability that die \(A\) is used is 32/33 when it is given that red turns up every time in first \(n\) throws, then find the value of \(n\).

There are 10 prizes, five A's, three B's, and two C's, placed in identical sealed envelopes for the top 10 contestants in a mathematics contest. The prizes are awarded by allowing winners to select an envelope at random from those remaining. When the 8th contestant goes to select the prize, the probability that the remaining three prizes are one A, one B and one C is

A point with coordinates (x, y) is chosen at random from the unit square. What is the probability that the point satisfies \(y^2 \leq x\)?

Three integers are chosen at random from the set of first 20 natural numbers. The chance that their product is a multiple of 3 is

\(A\) and \(B\) are two independent events such that \(P(A \cap B') = \frac{1}{5}\) and \(P(A' \cap B) = \frac{1}{6}\) then \(P(B)\) is equal to

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

An unbiased cubic die marked with 1, 2, 2, 3, 3, 3 is rolled 3 times. The probability of getting a total score of 4 or 6 is

For Problems 4–6: In an objective paper, there are two sections of 10 questions each. For 'section 1', each question has 5 options and only one option is correct and 'section 2' has 4 options with multiple answers and marks for a question in this section is awarded only if he ticks all correct answers. Marks for each question in 'section 1' is 1 and in 'section 2' is 3. (There is no negative marking.)The probability of getting a score less than 40 by answering all the questions by guessing in this paper is

\(AB\) and \(\overline{AB}\) are mutually exclusive and exhaustive events. If \(P(AB) = \frac{1}{25}\), then \(P(\overline{A} \cdot \overline{B}) = 1 - P(AB)\) equals:

If two events A and B are such that \(P(A) = 0.3\), \(P(B) = 0.4\) and \(P(A' \cap B') = 0.5\), then find the value of \(P(B/(A \cup B'))\).

A fair die is tossed repeatedly. \(A\) wins if it is 1 or 2 on two consecutive tosses and \(B\) wins if it is 3, 4, 5 or 6 on two consecutive tosses. The probability that \(A\) wins if the die is tossed indefinitely is

A dice is thrown three times and the sum of the thrown numbers is 15. Find the probability for which number 4 appears in first throw.

A bag contains 3 red, 4 black and 2 white balls. Three balls are drawn at random. What is the probability that the three balls have different colours?

On a Saturday night, 20% of all drivers in U.S.A. are under the influence of alcohol. The probability that a driver under the influence of alcohol will have an accident is 0.001. The probability that a sober driver will have an accident is 0.0001. If a car on a Saturday night smashed into a tree, the probability that the driver was under the influence of alcohol is

In a game show "Kaun Banega Dus Crore Pati" The host Mr. Kabir Khan gave the guest Mr Rajesh a choice of three doors: Behind one door is a new shining car; behind the others, nothing. Mr. Rajesh pick a door, say No. 1, and the host, Mr. Kabir Khan who knows what's behind the doors, opens another door, say No. 3, which has nothing. He then says to Mr Rajesh, "Do you want to pick door No. 2?" What he should do now to win the car?

If A and B are two events such that \(P(A \cup B) = P(A \cap B)\), then the incorrect statement amongst the following statements is:

The probability of event \(A\) and \(B\) occurring together is \(\frac{3}{2}\) and that of \(A\) and \(B\) occurring together is \(\frac{3}{10}\) then \(P(A^c) + P(B^c)\) is equal to

If a coin is tossed \(n\) times, then find the probability that the head appears odd number of times.

Find the probability that a randomly chosen three-digit number has exactly three factors.

One ticket is selected at random from 50 tickets numbered 00, 01, 02, …, 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, is

A coin is tossed \(n\) times. The probability of getting at least one head is at least \(\dfrac{99}{100}\). What is the minimum value of \(n\)?