Probability practice · Mixed difficulty

Let 20 distinct balls have been randomly distributed in to 4 distinct boxes, 5 into each. Let 'A' be the event that two specific balls have been put into a particular box. The probability of occurence of event ' $\mathrm{A}$ ' is :

A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P\left( {X \ge 6|X > 2} \right)$ is

Four fair dice $D_1, D_2, D_3$ and $D_4$ each having six faces numbered $1,2,3,4,5$ and 6 are rolled simultaneously. The probability that $D_4$ shows a number appearing on one of $D_1, D_2$ and $D_3$, is

An urn contains 4 white and 3 red balls. There balls are drawn with replacement from this urn. Then, the standard deviation of the number of red balls drawn is

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

There are three events ${E}_1, {E}_2$ and ${E}_3,$ one of which must, and only one can happen. The odds are 7 to 4 against ${E}_1$ and 5 to 3 against ${E}_2$. The odds against ${E}_3$ is

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. One fruit is picked out from each basket. The probability that the fruits are both apples or both oranges, is

There are 5 identical red balls and 3 identical yellow balls in a box. Two balls are drawn at random, then the probability that their colour is different is

From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability that the number obtained on the card is divisible by 6 or 8 but not by 24, is

Let $w$ be a complex cube root of unity with $w \neq 1$. 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 $w^{r_1}+w^{r_2}+w^{r_3}=0$ is

The probabilities of Ramesh using car or scooter or bus or train for going to office are respectively $\frac{1}{7}, \frac{3}{7}, \frac{2}{7}$ and $\frac{1}{7}$. The probabilities of his reaching the office late using these modes of transport are respectively $\frac{2}{9}, \frac{1}{9}, \frac{4}{9}$ and $\frac{1}{9}$, On one day Ramesh reaches his office on time. Then the probability that he used car on that day is

A pack of plying cards was found to contain only 51 cards. If the first 13 cards which are examined are all red, then the probability that the missing cards is black, is

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one in a random order till both the faulty machines are identified. Then, the probability that only two tests are needed, is

The probability of selecting integers $a \in \left[ { - 5,30} \right]\,$ such that ${x^2} + 2\left( {a + 4} \right)x\mathrm{ } - \mathrm{ }5a\mathrm{ } + \mathrm{ }64\mathrm{ } > \mathrm{ }0,\mathrm{ }$ for all $x \in R,$ is

A coin is tossed 3 times by 2 persons. The probability that both get equal number of heads, is

If $x=3^n,$ where $n$ is a positive integral value, then what is the probability that $x$ will have 3 at unit's place?

For a party 7 guests are invited by a husband and his wife. They sit in a row for dinner. The probability that the husband and his wife sit together, is

A die thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared at least once is

In an assembly of 4 persons the probability that at least 2 of them have the same birthday, is

If $A$ and $B$ are two events, such that $P\left( {A \cup B} \right) = \frac{3}{4},\;P\left( {A \cap B} \right) = \frac{1}{4},P\left( {\bar A} \right) = \frac{2}{3}$, then $P\left( {\bar A \cap B} \right)$ is equal to

Three identical dice are thrown together. The probability that distinct numbers appear on them, is

In eight throws of a die 1 or 3 is considered a success. Then, the standard deviation of success, is

If $A$ and $B$ are mutually exclusive events with $P\left( B \right) \ne 1$, then $P(A|\bar B)$ is equal to (Here, $\bar B$ is the complement of the event $B$)

Two numbers are selected randomly from the set $S=\{1,2,3,4,5,6\}$ without replacement one by one. The probability that the minimum of the two numbers is less than 4 is

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

Three groups $A, B, C$ are contesting for position on the Board of Directors of a company. The probabilities of their winning are $0.5,0.3,0.2$ respectively. If the group A wins, the probability of introducing a new product is 0.7 and the corresponding probabilities for group $B$ and $C$ are 0.6 and 0.5 respectively. The probability that the new product will be introduced, is

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let $X$ denote the random variable of numbers of aces obtained in the two drawn cards. Then $P(X=1)+P(X=2)$ equals

It is given the events $A$ and $B$ are such that $P\left( A \right) = \frac{1}{4},P\left( {A\mathrm{|}B} \right) = \frac{1}{2}$ and $P\left( {B\mathrm{|}A} \right) = \frac{2}{3}$. Then, $P\left( B \right)$ is

Two dice are tossed 6 times. Then the probability that 7 will show an exactly four of the tosses is

A pair of a dice thrown, if 5. appears on at least one of the dice, then the probability that the sum is 10 or greater, is