Probability practice · Mixed difficulty

The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chances of the events are

The probability that in a random arrangement of the letters of the word ‘UNIVERSITY’, the two $I'\,\mathrm{s}$ do not come together is

A die is thrown. If it shows a six, we draw a ball from a bag consisting 2 black balls and 6 white balls. If it does not show a six, then we toss a coin. Then, the sample space of this experiment consists of

Two non-negative integers are chosen at random. The probability that the sum of the square is divisible by 5 is

An unbiased die is rolled until two consecutive trails result in even numbered faces. The probability that exactly six trials are required to get two consecutive even numbered faces, is

Two numbers $x$ and $y$ are chosen at random from the set $\{1,2,3, \ldots, 30\}$. The probability that $x^2-y^2$ is divisible by 3 is

A person draws a card from a pack of playing cards, replaces it and shuffles the pack. He continues doing this until he draws a spade. The chance that he fail the first two times is

Three dice are thrown. The probability that the same number will appear on each of them, is

A coin is tossed 4 times. The probability that at least one head turns up, is

Probability is 0.45 that a dealer will sell at least 20 television sets during a day, and the probability is 0.74 that he will sell less than 24 televisions. The probability that he will sell $20,21,22$, or 23 televisions during the day, is

If $C$ and $D$ are two events such that $C \subset D$ and $P(D) \neq 0$, then the correct statement among the following is:

Let $0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$ and $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)\;P\left( B \right).$ Then,

From a bag containing 30 tickets numbered $1,2,3, \ldots 30,$ three tickets are selected at random without replacement. Suppose these tickets are arranged as $x_1 < x_2 < x_3$. The probability that $x_2=15$ and $x_3>20$, is

If $\mathrm{A}$ and $\mathrm{B}$ be two events such that $P(A)=\frac{3}{8}, P(B)=\frac{5}{8}$ and $P(A \cup B)=\frac{3}{4}$, then $P(A / B) \times P\left(A^{\prime} / B\right)$ is equal to

A six faced die is so biased that it is twice likely to show an even number as an odd number when thrown. If the die is thrown twice the probability that sum of the numbers is even is

Odds in favour of an event $A$ are 2 to 1 and odds in favour of $A \cup B$ are 3 to 1 . Consistent with this information the smallest and largest values for the probability of event $B$ are given by

The probability that at least one of the events $A$ and $B$ occurs is 0.7 and they occur simultaneously with probability 0.2. Then, $P\left( {\bar A} \right) + P\left( {\bar B} \right) =$

A person draws a card from a pack of 52 playing cards, replaces it and shuffles the pack. He continues doing this until he draws a spade, the chance that he will fail in the first two draws is

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is

A number $x$ is chosen from the set $A=\left\{33^n: n \in N\right\}.$ The probability that $x$ has 3 in units place, is

If two events $A$ and $B$ are such that $P\left( {{A^c}} \right) = 0.3,\;P\left( B \right) = 0.4$ and $P\left( {A \cap {B^c}} \right) = 0.5,$ then$\;P\left[ {\frac{B}{{\left( {A \cup {B^c}} \right)}}} \right]$ is equal to

A bag contains 5 red, 3 white and 2 black balls. If a ball is picked at random, the probability that it is red is

In a multiple choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed upto three chanes to answer the question, the probability that he will get marks in the question, is

If the probability density function of a random variable $X$ is $f\left( x \right) = \frac{x}{2}$ in $0 \le x \le 2,$ then $P\left( {\frac{{X > 15}}{{X > 1}}} \right)$ is equal to

Kiran is deciding whether to visit Russia or Ukrain for the holidays. She makes her decision by rolling a regular 6-sided die. If she gets 1 or 2 she goes to Ukrain, if she rolls a 3,4 or 5 she goes to Russia. If she rolls a 6 she rolls again, the probability that she goes to Ukrain is $p$, then $5 p$ is

Out of 40 consecutive integers, two are chosen at random, the probability that their sum is odd is

If $A\;\,\mathrm{and}\,\,B$ are arbitrary events, then

In a binomial distribution the mean is 15 and variance is 10. Then parameter $n$ is

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

If a die is thrown and a card is selected at random from a deck of 52 playing cards, then the probability of getting an even number on the die and a spade card is