Probability practice · Medium

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

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

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

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

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

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 $B$)

Let $S=\left\{M=\left[a_{i j}\right], a_{i j} \in\right\}\{0,1,2\},\{1 \leq i, j \leq 2\}$ be a sample space and $A\{M \in S: M$ is invertible $\}$ be an even. Then $P(A)$ is equal to

If $P\left( B \right) = \frac{3}{4},\;P\left( {A \cap B \cap \bar C} \right) = \frac{1}{3}\;$ and $P\left( {\bar A \cap B \cap \bar C} \right) = \frac{1}{3},\;$ then $P\left( {B \cap C} \right)$ is

If $n$ positive integers are taken at random and multiplied together, the probability that the last digit of the product is 2, 4, 6 or 8, is

In a bag there are 10 balls of either Black or White. Let $A_k$ be the event which contains exactly $k$ white balls and its probability is proportional to $k^2$ Now an another ball is drawn randomly. If $P(E)$ be the probability that the ball drawn is white and $P(E)=\frac{a}{b}$ Where $\operatorname{HCF}(a, b)=1$ then the value of $|\mathrm{a}-\mathrm{b}|$ equals

The number of times a die must be tossed to obtain a 6 at least once with probability exceeding 0.9 is at least

Let $E^{\prime}$ denote the complement of an event $E$. Let $E, F$, $G$ be pairwise independent events such that $P(G)>0$ and $P(E \cap F \cap G)=0$. Then $P\left(E^{\prime} \cap F^{\prime} / G\right)$ equals

A bag contains 5 white, 6 green, 7 blue, 8 red and 9 black balls. If 3 balls are drawn one by one with replacement after each drawing, then the probability that $3^{\text {rd}}$ ball drawn is green is, if it is known that second ball is green:

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters, TA, are visible. The probability that the letter has come from CALCUTTA is

A box contains 15 transistors, 5 of which are defective. An inspector takes out one transistor at random, examines it for defects and replaces it. After it has replaced another inspector does the same thing and then so does a third inspector. The probability that atleast one of the inspectors finds a defective transistor, is equal to

Throwing a biased die, a person will get 5 rupees if he throws the number 5 and will get 8 rupees for any
other number, then the expected income (in Rupees) per throw is (it is given that the number 5 will appear
5 times as frequently as any other number)

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

An anti-aircraft gun can take a maximum of four shots at any plane moving away from it. The probabilities of hitting the plane at the 1st, 2nd, 3rd and 4th shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that at least one shot hits the plane?

Let ${R}$ be the region containing all the points within or on the sides of an equilateral triangle. A point $P$ is randomly selected from the region $R$, the probability that $P$ lies near to internal angle bisector than any of the sides is equal to

$A$ 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 $P \cup Q=A$, is

Among 15 players, 8 are batsman and 7 are bowlers. The probability that a team is chosen of 6 batsman and 5 bowlers, is

If ${X}$ and ${Y}$ are the independent random variables ${B}\left(5, \frac{1}{2}\right)$ and ${B}\left(7, \frac{1}{2}\right)$, then ${P}({X}+{Y} \geq 1)=$

$A$ 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 $P \cap Q$ contain just one element, is

Minimum number of times a fair coin must be tossed so that the probability of getting atleast one head is more than 99% is

The probability that a randomly chosen 5-digit number is made from exactly two digits is

From first 100 natural numbers, 3 numbers are selected. If these numbers are in A.P., then the probability that these numbers are even is:

If $A$ and $B$ are two events such that $P\left( A \right) > 0$ and $P\left( B \right) \ne 1,$ then $P(\bar A|\bar B)$ is equal to

Two numbers are selected at random from the numbers $1,2, \cdots, n$. Let $p$ denote the probability that the difference between the first and second is not less than $m$ (where $0 < m< n$ ). If $n=25$ and $m=10$ then $5 p$ is equal