Probability Questions (959)

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

Let A and B be two events such that the probability that exactly one of them occurs is \(\frac{2}{5}\) and the probability that A or B occurs is \(\frac{1}{2}\), then the probability of both of them occur together is

In shuffling a pack of playing cards, four are accidently dropped. The probability that missing cards should be one from each suit, is

Football teams T1 and T2 have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T1 winning, drawing and losing a game against T2 are \(\frac{1}{2}\), \(\frac{1}{6}\) and \(\frac{1}{3}\) respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams T1 and T2 respectively, after two games. Find \(P(X > Y)\).

For Problems 10–12: Let \(S\) and \(T\) are two events defined on a sample space with probabilities \(P(S) = 0.5\), \(P(T) = 0.69\), \(P(S/T) = 0.5\).The value of \(P(S \text{ or } T)\) is

Given $P(A)=0.5$, $P(A\cup B)=0.8$. If $A$ and $B$ are mutually exclusive, $P(B)=p$. If $A$ and $B$ are independent, $P(B)=q$. Find $q/p$.

A six faced fair die is thrown until 1 comes. Then, the probability that 1 comes in even number of trials, is

On a chessboard small squares are either black or white, set alternately. Three pawn are placed at random on three squares of the chessboard. The probability that two are on the squares of the same colour, is

If any four numbers are selected and they are multiplied, then the probability that the last digit will be 1, 3, 5 or 7, is

A coin is tossed three times. Let \(A\): head on third toss, \(B\): heads on first two tosses. Find \(P(A/B)\).

We have \(A = \{4, 5, 6\}\), \(B = \{1, 2, 3, 4\}\). We have \(A \cup B = \{1, 2, 3, 4, 5, 6\} = S\) where \(S\) is the sample space of the experiment of throwing a die, so \(P(S) = 1\). Hence \(P(A \cup B) =\)?

Out of 21 tickets consecutively numbered, there are drawn at random. Find the probability that the numbers on them are in AP. If the probability is \(\frac{a}{b}\), then \((14a - b)\) is ……….

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is ………. (JEE Main 2020)

For three events A, B and C, if P(exactly one of A or B occurs) = P(exactly one of B or C occurs) = P(exactly one of C or A occurs) = \(\frac{1}{4}\) and P(all the three events occur simultaneously) = \(\frac{1}{16}\), then the probability that atleast one of the events occurs, is

From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a-b\geq10$ is $\dfrac{m}{n}$, $\gcd(m,n)=1$, then $m+n$ is equal to _____.

For independent events \(A_1, A_2, \ldots, A_n\), \(P(A_i) = \frac{1}{i+1}\), \(i = 1, 2, \ldots, n\). Then, the probability that none of the events will occur is

A point is selected at random from the interior of a circle. The probability that the point is closer to the centre than boundary of the circle is

The probability of getting exactly 2 heads in tossing a coin thrice is \(\left(\frac{1}{8}\right)^n\).

A box contains 3 white and 2 red balls. If we draw one ball and without replacing the first ball, the probability of drawing red ball in the second draw is

In throwing of a die, let A be the event 'an odd number turns up', B be the event 'a number divisible by 3 turns up' and C be the event 'a number \(\le 4\) turns up'. The probability that exactly two of A, B and C occur is \(\frac{a}{b}\), then \(a + b\) is ……….

Letters of the word MATHEMATICS are arranged in all the possible ways, and a word is selected randomly then the probability that letter $C$ is exactly between $S$ and $H$ is ______.

If four whole numbers taken at random are multiplied together, then find the probability that the last digit in the product is 1, 3, 7 or 9.

For Problems 16–18: Two fair dice are rolled. Let \(P(A_i) > 0\) denote the event that the sum of the faces of the dice is divisible by \(i\).Which one of the following events is most probable?

The probability of drawing one white ball and one green ball from the first urn is \(\frac{1}{5}\). The probability of drawing one white ball and one green ball from the second urn is \(\frac{1}{3}\). The probability of drawing one white ball and one green ball from the third urn is \(\frac{2}{11}\). Therefore, the probability that the third urn was chosen is \(\frac{a}{b}\). Find \(a + b\).

In a lottery there were 90 tickets numbered 1 to 90. Five tickets were drawn at random. The probability that two of the tickets drawn numbers 15 and 89, is

Fifteen coupons are numbered from 1 to 15. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9, is

If three numbers are selected from the set of the first 20 natural numbers, the probability that they are in GP, is

A and B are two events such that P(A) ≠ 0. P(B/A) if (i) A is a subset of B (ii) A ∩ B = ∅ are respectively

In a sequence of independent trials, the probability of success in one trial is \(\dfrac{1}{4}\). Find the probability that the second success takes place on or after the fourth trial.

At a telephone enquiry system, the number of phone calls regarding relevant enquiry follow Poisson's distribution with an average of 5 phone calls during 10 min time interval. The probability that there is atmost one phone call during a 10 min time period, is

A bag contains $10$ different balls. Five balls are drawn simultaneously and then replaced and then seven balls are drawn. The probability that exactly three balls are common to the two drawn is $p$ then the value of $10p$ is ______.

Let n ordinary fair dice are rolled once. The probability that at least one of the dice shows an odd number is \(\frac{31}{32}\) than 'n' is equal to:

If $$P(A) = \frac{3}{8}$$, $$P(B) = \frac{3}{8}$$ and $$P(A \cap B) = \frac{1}{4}$$, then $$P\left(\frac{A}{B}\right)$$ is equal to

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. One fruit is picked out from each basket. The probability that the fruits are both apples or both oranges, is

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

A die is thrown 4 times. Find the probability of getting at most two 6.

A three-digit number is selected at random from the set of all three-digit numbers. The probability that the number selected has all the three digits same is

Two fair dice are thrown till outcome is $12$. The probability that one has to do $20$ throws for this is ______.

If A and B are two events such that $$P(A) > 0$$ and $$P(B)

If the integers m and n are chosen at random from 1 to 100, then the probability that a number of the form 7n + 7m is divisible by 5 equals

One ticket is selected at random from 100 tickets numbered 00, 01, 02, ..., 99. Suppose \(A\) and \(B\) are the sum and product of the digit found on the ticket, respectively. Then \(P((A = 7)/(B = 0))\) is given by

A set contains 3n members. Let \(P_n\) be the probability that S is partitioned into 3 disjoint subsets with n members in each subset such that the three largest members of S are in different subsets. Then \(\lim_{n \to \infty} P_n =\)

Eight players $P_1, P_2, \ldots P_8$ play a knock out tournament. It is known that whenever the players $P_i$ and $P_j$ play, the player $P_i$ will win if $i < j$. Assuming that the players are paired at random in each round, then the probability that the player $P_4$ reaches the final is ______.

The probability that all the squares in any column are of same color and that of a row are of alternating color is

Let \(S\) be the set of all functions from the set \(\{1, 2, \ldots, 10\}\) to itself. One function is selected from \(S\), the probability that the selected function is one-one and onto is:

A die is thrown. Let A be the event that the number obtained is greater than 3 and B be the event that the number obtained is less than 5. Then, \(P(A \cup B)\) is

Logic on success events.

Box I contains 5 red and 2 blue balls, while box II contains 2 red and 6 blue balls. A fair coin is tossed. If it turns up head, a ball is drawn from box I, else a ball is drawn from box II. The probability that the ball drawn is from box I, if it is blue, is

An ordinary deck of $52$ playing cards is randomly divided into $4$ groups of $13$ cards each. The probability that each group has exactly $1$ jack?

Tickets 1-10 drawn without replacement. Prob |x - y| >= 4.