Probability

If $P(A) = 0.6, P(B) = 0.4$, and $P(A \cap B) = 0.2$, calculate $P(A'|B')$.

Two cards are drawn from a pack of 52 cards with replacement. What is the probability that the first is a spade and the second is a club?

A die is rolled. If $A = \{1, 2\}$ and $B = \{2, 3, 4\}$, find $P(A|B)$.

Given $P(A) = 0.2$, $P(B) = 0.8$, and $P(A \cup B) = 0.9$, find $P(A \cap B)$.

If $A$ and $B$ are independent events with $P(A) = 0.4$ and $P(B) = 0.5$, find $P(A' \cup B')$.

Two dice are rolled. Given that the sum of the numbers is 10, what is the probability that one of the dice shows a 4?

If $P(A \cup B) = 0.8$ and $P(A \cap B) = 0.3$, find $P(A') + P(B')$.

A box contains 3 black and 5 white balls. Two balls are drawn from the box one after another without replacement. Given that the second ball is white, what is the probability that the first ball is also white?

A coin is tossed until a head appears. Given that the first toss was a tail, what is the probability that the coin is tossed at least 4 times in total?

If $P(A) = 0.8, P(B) = 0.5$ and $P(B|A) = 0.4$, find the value of $P(A \cap B^c | A \cup B)$.

If $P(A) = 0.25$ and $P(B) = 0.2$ are independent events, find $P(A \cap B)$.

A bag contains 3 green and 4 yellow balls. Two balls are drawn with replacement. What is the probability that the first is green and the second is yellow?

If $P(A) = 0.3$ and $P(B) = 0.3$ are independent, find $P(A \cup B)$.

A fair coin is tossed and a fair die is rolled. Find the probability of getting a 'Tail' and a number divisible by 3 on the die.

Three independent students try to solve a riddle. The probabilities are $1/3, 1/4, 1/5$. What is the probability that the riddle is NOT solved by any of them?

If $A$ and $B$ are independent events with $P(A) = 0.4$ and $P(B) = 0.5$, find the probability that only event $B$ occurs.

A coin is tossed and a card is drawn from a 52-card deck. What is the probability of getting a 'Tail' and a 'Spade'?

Two balls are drawn with replacement from a box containing 3 red and 7 white balls. The probability that both are of the same color is:

The probability of a student passing in Physics is $0.3$ and in Chemistry is $0.4$. Assuming independence, the probability that the student passes in at least one subject is:

If two events $A$ and $B$ are independent, then $P(A \cap B')$ is equal to:

Box A contains $10$ bulbs of which $2$ are defective. Box B contains $10$ bulbs of which $1$ is defective. A box is chosen at random and a bulb is drawn. What is the probability the bulb is defective?

A die is rolled. If the number is even, Bag 1 ($3$ white, $2$ black) is chosen. If the number is odd, Bag 2 ($2$ white, $3$ black) is chosen. What is the probability of drawing a white ball?

Employee A handles $40\%$ of the files with an error rate of $1\%$. Employee B handles $60\%$ with an error rate of $2\%$. If an error is found, what is the probability it was caused by Employee B?

Box X contains $5$ red and $3$ black balls. Box Y contains $3$ red and $5$ black balls. One box is chosen at random and a ball is drawn. What is the probability it is red?

Person $A$ tells the truth $3$ out of $4$ times, and Person $B$ tells the truth $5$ out of $6$ times. They both state that a fair coin toss resulted in 'Heads'. What is the probability that it was actually 'Heads'?

In an exam, a student knows the answer with probability $0.6$, guesses with probability $0.3$, and copies from a neighbor with probability $0.1$. The probability of being correct is $1.0$ if he knows, $0.25$ if he guesses, and $0.5$ if he copies. If he answered correctly, what is the probability he actually knew the answer?

Bag 1 has $2$ red, $3$ white, and $4$ black balls. Bag 2 has $3$ red, $2$ white, and $2$ black balls. Bag 3 has $4$ red, $1$ white, and $3$ black balls. A bag is chosen and a red ball is drawn. Find the probability it came from Bag 3.

If a person has a specific disease, a diagnostic test shows a positive result in $95\%$ of cases. If the person does not have the disease, the test shows a positive result in $2\%$ of cases. If $0.5\%$ of the population has the disease, what is the probability that a person who tests positive actually has the disease?

Three urns contain: Urn $I$ ($1$ white, $2$ black), Urn $II$ ($2$ white, $1$ black), Urn $III$ ($2$ white, $2$ black). One urn is chosen at random and two balls are drawn without replacement. They are found to be one white and one black. What is the probability that Urn $III$ was chosen?

The probability that a new ship will get an engine design award is $0.25$, the probability that it will get an accommodation design award is $0.35$ and the probability that it will get both awards is $0.15$. Given that the ship has received the engine design award, what is the probability that it also received the accommodation design award?

If $E(X) = 4$, then $E(2X+3)$ is:

In a Binomial distribution, if $p=q$, then the value of $p$ is:

For a random variable $X$, $E(X^2) - [E(X)]^2$ calculates the:

The square of the standard deviation is called:

A random variable $X$ has the probability distribution: $X: 0, 1, 2$ $P(X): 0.1, k, 2k-0.1$\nFind the value of $k$.

A box has 10 items, 2 are defective. Three items are chosen with replacement. Let $X$ be the number of defectives. Find $P(X=1)$.

What is the variance of the number of heads when 100 fair coins are tossed?

Let $X$ be the number of failures before the first success in independent Bernoulli trials with success probability $p$. Find $E[X]$.

If $X$ follows $B(n, p)$ and $P(X=k)$ is the maximum probability, then $k$ must satisfy which of the following?

A continuous random variable $X$ has the PDF $f(x) = k \sin x$ for $0 \le x \le \pi$. Find $E[X]$.

If $E(X) = 7$ and $Var(X) = 0$, what is $P(X=7)$?

Which of the following is the correct formula for $Var(X)$?

If a random variable $X$ takes values $a$ and $b$ with probability $0.5$ each, what is the mean of $X$?

A discrete random variable has a probability mass function $P(X=x) = \frac{1}{k}$ for $x = 1, 2, 3, 4$. What is the value of $k$?

Given $E(X^2) = 25$ and $E(X) = 4$, find the value of $E((X-4)^2)$.

If $Var(X) = 4$, find the standard deviation of the random variable $Y = 5 - 2X$.

Let $P(X=x) = \frac{k}{x}$ for $x=1, 2, 3$. Find the expected value $E(X)$.

A random variable $X$ has the probability distribution $x=1, 2, 3, 4, 5$ with $P(X=x) = \frac{1}{5}$. Find $E(2X^2 - 3)$.

Two dice are thrown. If $X$ is the product of the numbers on the dice, find $E(X)$.

A discrete random variable $X$ has $P(X=x) = \frac{k}{x}$ for $x=1, 2, 3$. Find the variance $Var(X)$.