Probability

Two dice are thrown. If the sum of the numbers is 7, find the probability that one of the dice shows 3.

Given $P(A) = 0.6$, $P(B) = 0.2$ and $P(A|B) = 0.5$, calculate $P(B|A)$.

If $P(A) = P(B) = x$ and $P(A \cap B) = x^2$, then $A$ and $B$ are always:

A card is drawn from a deck. Let $E$ be the event that the card is a spade and $F$ be the event that it is an ace. Find $P(E|F)$.

$A$ and $B$ are independent events such that $P(A) = p$, $P(B) = 2p$. If the probability that exactly one of them occurs is $\frac{5}{9}$, find the value of $p$.

A fair die is rolled. If the number turned up is even, what is the probability that it is the number $4$?

If $P(A) = 0.7$, $P(B) = 0.4$ and $A$ and $B$ are independent events, find the value of $P(A \cap \bar{B})$.

The probability that a bullet hits a target is $1/3$. If $3$ bullets are fired, what is the probability that at least one hits the target given that at most two bullets hit?

If $A$ and $B$ are two events such that $P(A) \neq 0$, then $P(B \mid A) = 1$ implies which of the following?

A box contains $3$ red, $4$ blue, and $5$ green balls. Two balls are drawn one by one without replacement. Find the probability that the second ball is green given that the first ball was not green.

If $A$ and $B$ are independent events such that $P(A) = 1/2$ and $P(B) = 1/3$, then $P(A \cup B)$ is:

A card is drawn from a deck of 52. $E$: the card is a spade, $F$: the card is an ace. Are $E$ and $F$ independent events?

If $P(A) = 0.2$ and $P(B) = 0.6$, and $A$ and $B$ are independent, what is $P(A' \cap B')$?

In a village, $60\%$ of families own a cow and $30\%$ own a buffalo. $15\%$ own both. Are the events 'owning a cow' and 'owning a buffalo' independent?

A card is drawn from a well-shuffled deck of 52 cards. Let $E$ be the event 'the card is a spade' and $F$ be the event 'the card is an ace'. Are $E$ and $F$ independent?

Suppose we have two bags. Bag 1 contains 2 red and 3 black balls. Bag 2 contains 3 red and 4 black balls. A bag is selected at random and a ball is drawn. What is the total probability that the ball drawn is red?

The probability that a student passes in Physics is $0.7$ and in Chemistry is $0.4$. If the events are independent, what is the probability that he passes in at least one of these subjects?

Two cards are drawn from a pack of 52 cards without replacement. What is the probability that the second card is a heart?

A test for a disease is $95\%$ accurate for those who have it and $94\%$ accurate for those who do not. If $0.1\%$ of the population has the disease, what is the probability that a person actually has the disease given that the test result is positive?

If $P(A) = 0.4, P(B) = p$ and $P(A \cup B) = 0.6$. If $A$ and $B$ are independent, find $p$.

The mean of a random variable $X$ is 10. What is $E(2X + 5)$?

If $X$ represents the outcome of rolling a fair die, then $P(X \text{ is even})$ is:

A random variable $X$ has $P(X=1)=0.5$ and $P(X=2)=0.5$. Find $Var(X)$.

For a random variable $X$, $E(X)$ is also known as the:

A random variable $X$ has $P(X=x) = k\left(\frac{1}{2}\right)^x$ for $x=1, 2, 3$. Determine $k$.

A fair coin is tossed 5 times. Let $X$ be the number of tails. Find $E(X)$.

A random variable $X$ has $P(X=0) = a$ and $P(X=1) = ab$. If $a + ab = 1$ and $a = \frac{2}{3}$, find the value of $b$.

In a sequence of 4 independent trials, the probability of success is $1/4$. Let $X$ be the number of successes. Find $E(X^2)$.

Let $X \sim B(n, p)$. If $n=10$ and $E(X)=5$, find $P(X=5)$.

A random variable $X$ has $P(X=0)=0.2$, $P(X=1)=0.3$, $P(X=2)=0.5$. Find $E(X^3)$.

A random variable $X$ takes the values $1, 2, 3$ with probabilities $k, 2k, 3k$. The mean of $X$ is:

A random variable $X$ has a mean of 5. What is the mean of $Y = 2X - 3$?

A random variable $X$ takes values $1, 10, 100$ with probabilities $0.7, 0.2, 0.1$ respectively. The mean is:

Let $X$ be the number of face cards (Jack, Queen, King) obtained when drawing one card from a deck. The mean of $X$ is:

Let $X$ be a random variable with $P(X=0)=0.4$ and $P(X=1)=0.6$. Find $E(2X+1)$.

Two fair coins are tossed. Let $X$ be the number of heads. Find $E(X^2)$.

For a binomial distribution with $n=4$ and $p=0.25$, find the value of $E(X^2)$.

A shooter's probability of hitting a target is $0.75$. If he fires $5$ shots, the mean number of hits is:

A bag contains $5$ red and $2$ white balls. Three balls are drawn without replacement. Let $X$ be the number of white balls. Find $E(X)$.

If a random variable $X$ follows a binomial distribution with $n=5$ and $p=0.4$, find $E(X)$.

If $P(E) = 0.2$ and $P(F) = 0.5$, and $P(E|F) = 0.3$, find $P(E \cap F)$.

If $P(A|B) = 1$, then which of the following is true regarding $A \cap B$?

Given $P(A) = 0.4$, $P(B) = 0.7$, and $P(B|A) = 0.5$, find $P(A \cup B)$.

If $P(A \cap B) = P(A) \cdot P(B)$, then $P(A|B)$ is equal to:

A committee of two members is to be selected at random from a group of 3 men and 2 women. Let $E$ be the event that the committee contains at least one woman and $F$ be the event that the committee contains exactly one man. Find the value of $P(F \mid E)$.

Let $A$ and $B$ be two events such that $P(A) = \frac{2}{5}$, $P(B) = \frac{1}{3}$, and $P(A \cap B) = \frac{1}{5}$. Calculate the value of the conditional probability $P(A \cap B' \mid A \cup B)$.

Given $P(A) = 0.7$ and $P(B) = 0.4$. If $A$ and $B$ are independent, find $P(A \cup B | B)$.

Let $A, B$, and $C$ be three events such that $P(A) = 0.3, P(B) = 0.4$, and $P(C) = 0.5$. Suppose that the events are pairwise independent and the probability that all three events occur simultaneously is 0. Find the value of the conditional probability $P(A' \cap B' | C)$.

If $P(A) = 0.4, P(B) = 0.8$, and $P(B|A) = 0.6$, find $P(A \cap B' | A)$.

Let $P(A) = 0.2, P(B) = 0.3$. If $A$ and $B$ are independent, calculate $P(A \cap B | A \cup B)$.

For a partition of $S$, the probability of the union of all partition events is exactly equal to the probability of the ______.

If $E_1, E_2, \dots, E_n$ are mutually exclusive and exhaustive, what is $P(E_1 \cup E_2 \cup \dots \cup E_n)$?

Let $S$ be the set of integers. Do 'Positive Integers' and 'Negative Integers' form a partition of $S$?

Is the collection $\{\{1, 2\}, \{3, 4\}, \{5\}\}$ a partition of $S = \{1, 2, 3, 4, 5, 6\}$?

A bag contains 3 white and 2 black balls. Another bag contains 2 white and 3 black balls. One bag is selected at random and a ball is drawn. If the ball is white, what is the probability it was from the first bag?

If $E_1$ and $E_2$ partition $S$ such that $P(E_1) = 0.4$ and $P(E_2) = 0.6$, and $P(A|E_1) = 0.1, P(A|E_2) = 0.2$, what is $P(A)$?

A box contains 5 red and 10 black balls. A ball is drawn at random, its color is noted and then it is returned to the box. If the color was red, 2 additional red balls are added; if black, 2 additional black balls are added. Then a second ball is drawn. Find the probability that the second ball is red.

A person tosses a fair six-sided die. If the outcome is a prime number, he draws two cards at random without replacement from a well-shuffled deck of 52 cards. If the outcome is not a prime number, he draws only one card from the deck. Given that he has drawn at least one King, what is the probability that the die showed a prime number?

An urn contains 5 balls, each of which is either white or black. Assuming that all possible compositions of the urn (from 0 to 5 white balls) are equally likely, two balls are drawn at random without replacement and are found to be white. What is the probability that all the balls in the urn are white?

A digital signal transmitter sends binary digits '0' and '1' with probabilities $0.4$ and $0.6$ respectively. The signal passes through two independent repeaters in series. Each repeater has a probability of $0.2$ of flipping the digit (changing '0' to '1' or '1' to '0'). If the digit received at the end of the second repeater is '1', what is the probability that the digit originally sent was '1'?

A factory has two production lines. Line 1 produces 2000 units per day and Line 2 produces 3000 units per day. The percentage of defective units are 1% and 2% respectively. Find the probability that a unit selected from the day's production is defective.

An urn contains 3 white and 2 black balls. A second urn contains 2 white and 4 black balls. A ball is drawn from the first urn and placed into the second. Then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is black.

A student answers a question. The probability that he knows the answer is 0.7 and that he guesses is 0.3. If he guesses, the probability of being correct is 1/4. Find the probability that his answer is correct.

A bag contains 2 red, 3 green and 2 blue balls. Another bag contains 3 red, 2 green and 4 blue balls. A bag is chosen at random and a ball is drawn. Find the probability that it is blue.

A logistics company uses three delivery services $L_1, L_2$, and $L_3$ for its operations. $50\%$ of the shipments are handled by $L_1$, $30\%$ by $L_2$, and $20\%$ by $L_3$. The probabilities of a shipment being delayed by $L_1, L_2$, and $L_3$ are $0.02, 0.04$, and $0.05$ respectively. If a shipment is selected at random, what is the probability that it arrives without any delay?

A factory has three machines $A, B, C$ producing $1000, 2000, 3000$ bolts per day respectively. $A$ produces $1\%$ defective bolts, $B$ produces $1.5\%$ and $C$ produces $2\%$ defective bolts. A bolt is chosen at random and found to be defective. Find the probability it was produced by machine $C$.

If a card is drawn from a well-shuffled pack of 52 cards, what is the probability that it is either a king or a queen?

In a certain city, 40% of the people have brown hair, 25% have brown eyes, and 15% have both. If a person is selected at random, what is the probability that the person has brown hair given that he has brown eyes?

An urn contains 5 white and 10 black balls. A ball is drawn at random, its color is noted and it is returned to the urn with 5 additional balls of the same color. What is the probability that the second ball drawn is white?

A box contains 100 tickets numbered 1 to 100. Two tickets are chosen at random. It is given that the maximum number on the two tickets is not more than 10. What is the probability that the minimum number on them is 5?