Probability Questions (959)

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

There are two red, two blue, two white, and certain number (greater than 0) of green socks in a drawer. If two socks are taken at random from the drawer without replacement, the probability that they are of the same color is 1/5, then the number of green socks are ________.

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 \(i\).For which one of the following pairs \((i, j)\) are the events \(A_i\) and \(A_j\) independent?

Out of 6 pairs of distinct gloves 8 gloves are randomly selected, then the probability that there exist exactly 2 pairs in it is :

If $a$ and $b$ are chosen randomly from the set consisting of numbers $1, 2, 3, 4, 5, 6$ with replacement. If the probability that $\lim_{x \to 0} \left(\frac{a^x + b^x}{2}\right)^{\frac{1}{x}} = 6$ is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - p = \ldots\ldots\ldots\ldots\ldots\ldots$

Dialing a telephone number an old man forgets the last two digits remembering only that these are different dialed at random. The probability that the number is dialed correctly is

The sum of the digits of a seven-digit number is 59. Find the probability that this number is divisible by 11.

For three events \(A\), \(B\) and \(C\),\(P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs})\)\(P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \dfrac{1}{4}\) and\(P(\text{All the three events occur simultaneously}) = \dfrac{1}{16}\).Then the probability that at least one of the events occurs, is

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$. If $p : q = m : n$, where $m$ and $n$ are coprime, then $m + n$ is equal to ___.

If \(a^2 + 4b^2 + 4c^2 - 2ab - 4bc - 2ac = 0\), then the number of ordered triplets \((a, b, c)\) with \(a, b, c \in \{1, 2, 3, 4, 5, 6\}\) satisfying the equation is 3, i.e., \((2,1,1), (4,2,2), (6,3,3)\). Two points \((2,1,1)\) and \((4,2,2)\) lie inside a given tetrahedron. If the required probability is \(\dfrac{2}{3} = \dfrac{5}{\lambda}\), then \(\lambda =\)

A bag has 10 balls. Six balls are drawn in an attempt and replaced. Then another draw of 5 balls is made from the bag. The probability that exactly two balls are common to both the draw is

An insurance company insured 2000 scooter drivers, 4000 car drivers, and 6000 truck drivers. The probability of accidents are 0.01, 0.03, and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

A letter is known to have come from CHENNAI, JAIPUR, NAINITAL, DUBAI and MUMBAI. On the post mark only two consecutive letters AI are legible. Then the probability that it come from MUMBAI, is

An urn contains 6 black balls and unknown number \((\leq 6)\) of white balls. Three balls are drawn successively and not replaced and are all found to be white. Prove that the chance that a black ball will be drawn in the next draw is \(\dfrac{677}{909}\).Let \(E_3\) = the event of the urn containing at least 3 white balls. Find the probability that a black ball will be drawn in the next draw given that 3 whites have already been drawn.

260. The probability of occurrence of a multiple of 2 on one dice and a multiple of 3 on the other dice if both are thrown together, is:

A problem in mathematics is given to three students A, B and C and their respective probability of solving the problem is 1/2, 1/3 and 1/4. Probability that the problem is solved, is

Five different digits from the set of numbers \(\{1, 2, 3, 4, 5, 6, 7\}\) are written in random order. Find the probability that the five-digit number thus formed is divisible by 9.

Let \(\omega\) be a complex cube root of unity with \(\omega \neq 1\). A fair die is thrown three times. If \(r_1\), \(r_2\) and \(r_3\) are the numbers obtained on the die, then the probability that \(\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0\) is

For mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\) and \(P(A \cap B) = 0\). Consider the experiment of throwing a die. Let \(A\) = event that the number obtained is odd and \(B\) = event that the number obtained is even. Which of the following statements is correct?

The probability that exactly one of the two events \(A, B\) occurs is

Cards are drawn one-by-one at random from a well-shuffled pack of 52 playing cards until 2 aces are obtained from the first time. The probability that 18 draws are required for this is

Four candidates A, B, C and D have applied for the post in government office. If A is twice as likely to be selected as B, and B and C are given about the same chances of being selected, while C is twice as likely to be selected as D, what are the probabilities that(i) C will be selected?(ii) A will not be selected?

Two numbers \(a, b\) are chosen from the set of integers 1, 2, 3, …, 39. Then probability that the equation \(7a - 9b = 0\) is satisfied is

Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?

The letters of the word ASSASSIN are written down at random in a row. The probability that no two S occur together is

A random variable $X$ takes values $0,1,2,3$ with probabilities $\dfrac{2a+1}{30}$, $\dfrac{8a-1}{30}$, $\dfrac{4a+1}{30}$, $b$ respectively, where $a,b\in\mathbb{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2+\mu^2=2$. Then $\dfrac{a}{b}$ is equal to:

The probability distribution of a random variable $X$ is given below. If $E(X)=\dfrac{263}{15}$, then $P(X<20)$ is equal to:

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.)If a candidate attempts only two questions by guessing, one from 'section 1' and one from 'section 2', the probability that he scores in both questions is

Each of the \(n\) urns contains 4 white and 6 black balls. The \((n+1)\)th urn contains 5 white and 5 black balls. One of the \(n+1\) urns is chosen at random and two balls are drawn from it without replacement. Both the balls turn out to be black. If the probability that the \((n+1)\)th urn was chosen to draw the balls is \(1/16\), then find the value of \(n\).

If \(A\) and \(B\) are any two events such that \(P(A) = 2/5\) and \(P(A \cap B) = 3/20\), then the conditional probability, \(P(A \mid (A' \cup B'))\), where \(A'\) denotes the complement of \(A\), is equal to

The probability that \(A\) speaks truth is \(4/5\), while this probability for \(B\) is \(3/4\). The probability that they contradict each other when asked to speak on a fact is

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

Let \(P(A)\) be the probability of at least two girls, and \(P(B)\) be the probability of all girls in a family with 4 children (probability of a girl = \(\dfrac{1}{4}\)). Find the required conditional probability \(\dfrac{P(B \cap A)}{P(A)}\).

Consider the following assignments of probabilities for outcomes of sample space \(S = \{1, 2, 3, 4, 5, 6, 7, 8\}\).Number (X)12345678Probability P(X)0.150.230.120.100.200.080.070.05Find the probability that (b) \(X\) is a number greater than 4.

A bag contains 10 balls out of which $k$ are red and $(10-k)$ are black, where $0\leq k\leq10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

A pair of unbiased dice is rolled together till a sum of either 5 or 7 is obtained. The probability that 5 comes before 7 is

The mean and the variance of a binomial distribution are 4 and 2, respectively. Then the probability of 2 successes is

A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the 4th time on the 7th draw is

Let \(A\) and \(E\) be any two events with positive probabilities.Statement-1: \(P(E/A) \geq P(A/E)P(E)\)Statement-2: \(P(A/E) \geq P(A \cap E)\).

Two persons \(A\) and \(B\) throw two dice each. If \(A\) throws a sum of 9 then the probability of \(Y\) throwing a sum greater than that of \(X\) is

For a binomial distribution, if \(P(X=2) = P(X=3)\) and \({}^nC_r(1-p)^{n-r}p^r = P(X=r)\), find \(E(X)\).

If \(A\) and \(B\) are two independent events then \(P(A \cap B) + P(A)P(B) = P(A)\).

A and B are two independent events. C is an event in which exactly one of A or B occurs. Prove that \[P(C) \geq P(A \cup B) \cdot P(\bar{A} \cap \bar{B}).\]

A coin is tossed three times in succession. If \(E\) is the event that there are at least two heads and \(F\) is the event in which first throw is a head, then find \(P(E/F)\).

Let gain/loss = \(W \times 100 + LW(-50 + 100) + L^2W(-50 - 50 + 100) + L^3(-150)\)where \(W\) is probability that outcome is 5 or 6 and \(L\) is probability that outcome is 1, 2, 3, 4. The expected gain/loss is:

Let two fair six-faced dice \(A\) and \(B\) be thrown simultaneously. If \(E_1\) is the event that die \(A\) shows up four, \(E_2\) is the event that die \(B\) shows up two and \(E_3\) is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT True?

Consider the following assignments of probabilities for outcomes of sample space \(S = \{1, 2, 3, 4, 5, 6, 7, 8\}\).Number (X)12345678Probability P(X)0.150.230.120.100.200.080.070.05Find the probability that (a) \(X\) is a prime number.

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is:

The probability that a particular day in the month of July is a rainy day is 3/4. Two person whose credibility are 4/5 and 2/3, respectively, claim that 15th July was a rainy day. Find the probability that it was really a rainy day.

Let A = {0, 5, 10, 15, ..., 195}. Let B be any subset of A with at least 15 elements. What is the probability that B has at least one pair of elements whose sum is divisible by 15?