Probability practice · Medium

Let $M$ be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space $S=\left\{x \in \mathbb{Z}: x(66-x) \geq \frac{5}{9} M\right\}$ and the event $A=\{x \in S: x$ is a multiple of 3$\}$. Then $P(A)$ is equal to

The probability of getting a total of at least 6 in the simultaneously throw of three dice is

A and B throw a die alternatively. If A starts throwing then the probability that he throws 5 before B throws 3 or 6 is

Probability that Ajay will not go to office is $\frac{1}{5}$ and probability that Ajay and Vijay will go to office is $\frac{2}{7}$, if their visits to office is independent of each other then find probability that Ajay will go to the office but Vijay will not go, is

Three people each flip two fair coins. The probability that exactly two of the people flipped one head and one tail, is

Of a total of 600 bolts, 20% are too large and 10% are too small. The remainder are considered to be suitable. If a bolt is selected at random, the probability that it will be suitable is

A sample of a 4 times is drawn at a random without replacement from a lot of 10 items containing 3 defectives. If $x$ denotes the number of defective items in the sample, then $P(0 < x < 3)$ is equal to

A five digit number if formed by writing the digits 1, 2, 3, 4, 5 in a random order without repetitions. Then, the probability that the number is divisible by 4 is

If two events $A$ and $B$ are such that $P\left( {{A^c}} \right) = 0.3,P\left( B \right) = 0.4,P\left( {A \cap {B^c}} \right) = 0.5,$ then $P\left( {B/A \cup {B^c}} \right) =$

A party is thrown for families having boys, girls, ladies and gentleman. When one member is randomly selected, odds against being a boy is $\frac{27}{5}$, odds in favour of being a girl is $\frac{7}{25}$ and probability of being a lady is $\frac{11}{32}$, then the odds in favour of the chosen person being a gentleman, is

If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn 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 $B$ shows up two and $E_3$ is the event that the sum of the numbers on both dice is odd, then which the following statements is not true?

$3 \%$ of the electric bulbs manufactured by a company are defective. Using Poisson distribution (approximation), the probability that a sample of 100 bulbs will contain exactly one defective, is

(Given antilog $(.1742)=1.5)$

Suppose that a die (with faces marked 1 to 6) is loaded in such a manner that for $K = 1,2,3, \ldots .,6$ the probability of the face marked $K$ turning up when die is tossed is proportional to $K$. The probability of the event that the outcome of a toss of the die will be an even number, is equal to

An integer is chosen at random from the integers $1,2,3, \ldots, 50$. The probability that the chosen integer is a multiple of atleast one of 4,6 and 7 is

If $X$ is a binomial variate with parameters $n$ and $p$, where $0 < p < 1$ such that $\frac{P(X=r)}{P(X=n-r)}$ is independent of $n$ and $r$, then $p$ equals

If $\mathrm{X}$ follows a binomial distribution with parameters $\mathrm{n}=100$ and $\mathrm{p}=1 / 3$, find $\mathrm{r}$ for which $\mathrm{P}(\mathrm{X}=\mathrm{r})$ is maximum

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and second draw gives all black balls is

For any two independent events ${E_1}$ and ${E_2},\;P\left\{ {\left( {{E_1} \cup {E_2}} \right) \cap \left( {{{\bar E}_1}} \right) \cap \left( {{{\bar E}_2}} \right)} \right\}$

A bag contains 5 white and 3 black balls and 4 balls are successively drawn out and not replaced. The probability that they are alternately of different colours, is

Two integers $r$ and $s$ are drawn one at a time without replacement from the set $1,2, \cdots, n$. If $p_k=P(r \leq k \mid s \leq k)$, then $4 p_7$ is if $n=25$.

Bag A contains 4 green and 3 red balls and Bag $\mathrm{B}$ contains 4 red and 3 green balls. One bag is taken at random and a ball is noted it is green. The probability that it comes from $\mathrm{Bag} \,\, \mathrm{B}$ is -

Let 0.8 the probability that a man aged 90 years will die in a year. If $p$ is the probability that out of 4 men $A_1, A_2, A_3$ and $A_4$ each aged $90, A_1$ will die in a year and will be the first to die, find $10000 p-2490$ is

A bag contains 8 red and 7 black balls. Two balls are drawn at random. The probability that both the balls are of the same colour, is

Two dice are rolled. If the probability of getting different numbers on the these dice is $\frac{p}{q}$ where $p, q$ are relatively prime then $q-p$ is

One percent of the population suffers from a certain disease. There is blood test for this disease, and it is $99 \%$ accurate, in other words, the probability that it gives the correct answer is 0.99 , regardless of whether the person is sick or healthy. A person takes the blood test, and the result says that he has the disease. The probability that he actually has the disease, is

A bag contains 50 tickets numbered 1, 2, 3, …, 50 of which five are drawn at random and arranged in ascending order of magnitude $\left( {{x_1} < {x_2} < {x_3} < {x_4} < {x_5}} \right).$ The probability that ${x_3} = 30,$ is

Three letters are written to there different persons and addresses on the three envelopes are also written. Without looking at the addresses, the letters are kept in these envelopes. The probability that all the letters are not placed into their right envelopes is

If $A$ and $B$ are independent events and $P\left( C \right) = 0,$ then

10 mangoes are to be distributed among 5 persons. The probability that at least one of them will receive none, is