Probability

If ' $a$ ' is an integer lying in $[-5,30]$, then the probability that the graph of $y=x^2+2(a+4) x-5 a+64$ lies above the $x-$ axis is

If $A_1, A_2, A_3, \ldots, A_{1006}$ be independent events such that $P\left(A_l\right)=\frac{1}{2 i},(i=1,2, \ldots, 1006)$ and the probability that none of the events occurs be $\frac{\alpha!}{2^a(\beta!)^2}$ ，then

Consider the sequence of numbers $(1,2,3, \ldots \ldots, 13)$. A person choose three numbers at random from the sequence. The probability that the chosen three number form an A.P. is

A mapping is selected at random from all mappings $f: A \rightarrow A$, where set $A=\{1,2,3 \ldots, n\}$. If the probability that the mapping is injective is $\frac{3}{32}$, then the value of $n$ is

Four natural numbers selected at random are multiplied together, then the probability that the digit in the unit's place in the product be $1,3,7$ or 9 is

Three numbers are chosen at random without replacement from $\{1,2, \ldots 10\}$. The probability that the minimum of the chosen numbers is 3 or their maximum is 7 , is

The probability that a non-leap year selected at random will have 53 Sundays or 53 Saturdays is

If $E$ and $F$ are two independent events with $P(E)=0.3$ and $P(E \cup F)=0.5$, then $P(E / F)-P(F / E)$ equals

A biased coin with probability \(\mathrm{p}(0<\mathrm{p}<1)\) of getting head is tossed until a head appears for the first time. If the probability that the number of tosses required is even is \(\frac{2}{5}\), then \(\mathrm{p}=\)

Two integers \(\mathrm{r}\) and \(\mathrm{s}\) are drawn one at a time without replacement from the set \(\{1,2, \ldots, \mathrm{n}\}\). Then \(\mathrm{P}(\mathrm{r} \leq \mathrm{k} / \mathrm{s} \leq \mathrm{k})=\)
(k is an integer < n)

Two smallest squares are chosen one by one on a chess board. The probability that they have a side in common is

Let S be the sample space of the random experiment of throwing simultaneously two unbiased dice and \(\mathrm{E_k=\{(a,b)\in S:ab=k\}}\). If \(\mathrm{p_k=P(E_k)}\), then the correct among the following is :

Let A and B are two independent events. The probability that both A and B happen is \({1 \over {12}}\) and probability that neither A and B happen is \({1 \over 2}\). Then

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is

A, B, C are mutually exclusive events such that \(P(A) = {{3x + 1} \over 3}\), \(P(B) = {{1 - x} \over 4}\) and \(P(C) = {{1 - 2x} \over 2}\). Then the set of possible values of x are in

Four persons A, B, C and D throw and unbiased die, turn by turn, in succession till one gets an even number and win the game. What is the probability that A wins the game if A begins?

Four persons A, B, C and D throw an unbiased die, turn by turn, in succession till one gets an even number and win the game. What is the probability that A wins if A begins?

A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire to have more than 50% chance of hitting it at least once, is

A and B are independent events. The probability that both A and B occur is \({1 \over {20}}\) and the probability that neither of them occurs is \({3 \over {5}}\). The probability of occurrence of A is

If X is a random variable such that \(\sigma\)(X) = 2.6, then \(\sigma\)(1 \(-\) 4X) is equal to

A problem in mathematics is given to 4 students whose chances of solving individually are \({{1 \over 2}}\), \({{1 \over 3}}\), \({{1 \over 4}}\) and \({{1 \over 5}}\). The probability that the problem will be solved at least by one student...

A student appears for tests I, II and III. The student is successful if he passes in tests I, II or I, III. The probabilities of the student passing in tests I, II and III are respectively p, q and 1/2. If the probability of the student to ...

In order to get a head at least once with probability \(\ge\) 0.9, the minimum number of times a unbiased coin needs to be tossed is

The probability that a non-leap year selected at random will have 53 Sunday is

If A, B are two events such that P(A \(\cup\) B) \(\ge\) \({3 \over 4}\) and \({1 \over 8}\) \(\le\) P (A \(\cap\) B) \(\le\) \({3 \over 8}\), then

Let A and B be two events such that P(A \(\cap\) B) = \({1 \over 6}\), P(A \(\cup\) B) = \({31 \over 45}\) and P(\(\overline B\)) = \({7 \over 10}\), then

In a group of 14 males and 6 females. 8 and 3 of the males and females, respectively are aged above 40 yr. The probability that a person selected at random from the group is aged above 40 yr given that the selected person is a female, is

Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then, the probability that the player gets all distinct cards is

A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is

4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is

The probability that at least one of A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, then P(A') + P(B') is

Two dice are tossed once. The probability of getting an even number at the first die or a a total of 8 is

Three numbers are chosen at random from 1 to 20. The probability that they are consecutive is

A and B are two independent events such that P(A \(\cup\) B') = 0.8 and P(A) = 0.3. Then P(B) is

If an unbiased coin is tossed n times. Find the probability that head appears an odd number of times.

A person draws out two balls successively from a bag containing 6 red and 4 white balls. The probability that at least one of them will be red is