Probability

A spinner is divided into $8$ equal sections numbered $1$ to $8$. What is the theoretical probability of landing on an even number?

In a class of $25$ students, $15$ have a pet. What is the relative frequency of students who have a pet?

If you roll a fair six-sided die $600$ times, how many times would you expect to roll a '$4$'?

What is the theoretical probability of drawing an Ace from a standard deck of $52$ cards?

A production line has a probability of 0.045 for minor flaws. In a random sample of 2,000 items, what is the expected number of items with no minor flaws?

In 1,000 rolls of a die, the number 1 appeared 180 times. If the die is fair, the theoretical probability is $1/6$. What is the absolute difference between the relative frequency and the theoretical probability (to 4 decimal places)?

A jar contains 500 beans. 60 are black. A student takes a sample of 50 beans. How many black beans should he expect to see in the sample if the sample is representative?

A bag contains 10 coins. 3 are fair and 7 are biased with $P(H) = 0.6$. A coin is chosen at random and tossed. If the result is heads, what is the probability the coin was fair?

A factory produces components where 1% are defective. If a random sample of 5 components is taken, what is the probability that none are defective (to 4 decimal places)?

Events $X$ and $Y$ are such that $P(X) = 0.5$, $P(Y | X) = 0.2$, and $P(Y | X') = 0.4$. Find $P(X | Y)$.

The probability of entering a correct password on the first try is $0.9$. What is the probability of failing to enter it correctly on two consecutive independent attempts?

A basketball player makes a free throw with probability $0.7$. If they take two independent shots, what is the probability they make exactly one shot?

A bag contains $5$ yellow and $5$ green balls. Two balls are drawn without replacement. What is the probability that they are the same color?

A commuter can take a train (prob $0.7$) or a bus (prob $0.3$). The train is late $5\%$ of the time, and the bus is late $10\%$ of the time. What is the probability that the commuter takes the bus and it is late?

In a game of archery, the probability of hitting the bullseye is $0.4$. If an archer takes two shots, what is the probability they hit the bullseye exactly once?

Two marbles are drawn without replacement from a bag containing $3$ red and $7$ black marbles. Find the probability that the first is red and the second is black.

A student takes a bus that is late $20\%$ of the time. If the bus is late, the student is late for class $70\%$ of the time. If the bus is on time, the student is late for class only $10\%$ of the time. What is the probability the student is on time for class?

A box contains 10 light bulbs, 2 of which are defective. Bulbs are tested one by one without replacement until both defective bulbs are found. What is the probability that the process stops at the 3rd test?

Bag A has 4 white and 2 black balls. Bag B has 3 white and 3 black. A ball is drawn from A and moved to B, then a ball is drawn from B and moved to A. Find the probability that Bag A now contains 4 white and 2 black balls.

Two independent events A and B have probabilities $P(A) = x$ and $P(B) = x+0.2$. If the probability that at least one of them occurs is 0.88, find $x$.

If $P(A') = 0.7$ and $P(B') = 0.8$, and $A$ and $B$ are mutually exclusive, find $P(A \cup B)$.

Two independent events $A$ and $B$ have probabilities $P(A) = \frac{1}{5}$ and $P(B) = \frac{1}{4}$. Find $P(A \cap B)$.

If $P(A) = 0.05, P(B) = 0.15,$ and $P(C) = 0.2$ are mutually exclusive, find $P(A \cup B \cup C)$.

Player A and Player B play independent games. If $P(\text{A wins}) = 0.3$ and $P(\text{B wins}) = 0.3$, what is the probability they both win?

If $P(A) = 0.5, P(B) = 0.3,$ and $P(A \cap B) = 0.15$, find $P(A \cup B)$.

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

If $P(A) = 0.1$ and $P(B) = 0.9$, and they are mutually exclusive, find $P(A \cup B)$.

If $A, B, C$ are independent events with $P(A)=0.5, P(B)=0.6, P(C)=0.8$, find $P(A \cap B \cap C)$.

Events $A$ and $B$ are independent with $P(A) = 1/3$ and $P(B) = 1/4$. Find $P(A' \cap B')$.

Two events $A$ and $B$ are independent. $P(A) = 0.4$ and $P(B) = 0.5$. Find $P(A' \cup B)$.

Suppose $P(A) = 0.25$, $P(B) = 0.4$, and $P(A \cap B) = 0.1$. Find $P(A|B)$.

A bag contains 10 marbles: 3 red and 7 blue. Two marbles are drawn without replacement. What is the probability that the second is red, given that the first was red?

If $P(A) = 0.5$ and $P(B|A) = 0.8$, calculate $P(A \cap B)$.

A fair six-sided die is rolled. What is the probability that the outcome is greater than 3, given that it is an even number?

Given $P(A) = 0.5$, $P(B) = 0.4$, and $P(A|B) + P(B|A) = 1.3$, what is $P(A \cap B)$?

If $P(A) = 0.3$, $P(B) = 0.4$, and $P(A \cap B) = 0.1$, find $P(B' | A')$.

A biased coin is such that $P(H) = 0.6$. If it is tossed twice, find the probability that the first toss was a head given that at least one head was obtained.

If $P(A) = 0.5, P(B) = 0.6,$ and $P(A \cap B) = 0.2$, what is $P(A' | B')$?

An integer $x$ is selected from $1$ to $20$. Given $x$ is even, find the probability it is a multiple of 3.

In a group of 50 people, 20 like tea, 15 like coffee, and 5 like both. Given that a person likes at least one of the two, find the probability they like both.