Probability practice · Mixed difficulty

If $X$ follows a binomial distribution with parameters $n = 100$ and $P = \frac{1}{3}$, then $P\left( {X = r} \right)$ is maximum when $r$ is equal to

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

A die is thrown twice and the sum of the numbers appearing is observed to be 6 . The conditional probability that the number 4 has appeared at least once, is

Two dice are thrown together. If the numbers appearing on the two dice are different, then what is the probability that the sum is 6?

A coin is tossed $n$ times the probability of getting head at least once is greater than 0.8. Then the least value of such $n$ 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

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event.

Given below are two statements:

(S1) : If $P(A)=0$, then $A=\varnothing$

(S2) : If $P(A)=1$, then $A=\Omega$

Then

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

A bag contains 4 tickets numbered 1, 2, 3, 4 and another bag contains 6 tickets numbered 2, 4, 6, 7, 8, 9. One bag is chosen and a ticket is drawn. The probability that the ticket bears the number 4 is

If $A$ and $B$ are mutually exclusive events with $P(B) \neq 1$, then $P(A / \bar{B})=$

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

If $A$ and $B$ are independent events such that $P\left( A \right) > 0,P\left( B \right) > 0,$ then

$A$ and $B$ are two events such that odds against $A$ are $2: 1$ odds in favour of $A \cup B$ are $3: 1$. If $x \leq P(B) \leq y$, then the ordered pair $(x, y)$ is

A coin is tossed $2 n$ times. The chance that the number of times one gets head is not equal to the number of times one gets tail, is

$A$ is a set containing $n$ elements. A subset $P$ of $A$ is chosen at random. The set $A$ is reconstructed by replacing the elements of $P$. A subset $Q$ is again chosen at random. The probability that $P$ and $Q$ have equal number of elements, is

A company has two plants A and B to manufacture motorcycles. $60 \%$ motorcycles are manufactured at plant A and the remaining are manufactured at plant B. $80 \%$ of the motorcycles manufactured at plant A are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then $126 \mathrm{p}$ is

It is $8: 5$ against a person who is now 40 years old living till he is 70 and $4: 3$ against a person now 50 living till he is 80 . The probability that at least one of these persons will be alive 30 years hence, is

Three coins are tossed together, then the probability of getting at least one head is

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

Out of $3 n$ consecutive integers, three are selected at random. Find the probability that their sum is divisible by 3 , is

Two events $A$ and $B$ are such that $P(B)=0.55$ and $P\left(A B^{\prime}\right)=0.15$. The probability of occurrence of at least one of event is

A bag contains 3 red and 7 black balls. Two balls are selected at random without replacement. If the second selected ball is given to be red, the Probability that the first selected ball is also red, is

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required. Then, the probability that $X=3$ equals

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

A bag contains 5 red and 3 bluc balls. If 3 balls are drawn at random without replacement, then the probability of getting exactly one red ball is

100 Students appeared for two examinations, 60 passed the first, 50 passed the second and 30 passed both. The Probability that a student selected at random has failed in both examinations, 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 -

A number is chosen at random among the first 120 natural numbers. The probability of the number chosen being a multiple of 5 or 15 is