🎲 probability

1. **State the problem:** We need to find the conditional probability that a card drawn from a standard 52-card deck is black, given that it is a spade. 2. **Recall the formula for

1. **State the problem:** We want to find the conditional probability that a card drawn is a spade, given that it is an ace. 2. **Recall the formula for conditional probability:**

1. **State the problem:** We have 5000 tickets sold at 1 each for a raffle. Prizes are:

1. **State the problem:** We want to find the expected value $E(X)$ of the amount won on a single raffle ticket when 5000 tickets are sold and prizes are distributed as follows: 1

1. **State the problem:** We have 5000 tickets sold at 1 each for a charity raffle. Prizes are: 1 prize of 800, 3 prizes of 300, 5 prizes of 10, and 20 prizes of 5. We want to find

1. **State the problem:** A card is drawn from a standard 52-card deck. If the card is a king, you win 30; otherwise, you lose 4. We want to find the expected value $E(X)$ of the a

1. **State the problem:** You pay 5 dollars to play a game where a fair six-sided die is rolled. You receive a payout equal to the number of dots on the die face. The net gain is t

1. **State the problem:** A fair coin is flipped. If a head turns up, you win 3. If a tail turns up, you lose 3. We want to find the expected value of the game and determine if the

1. **State the problem:** We toss 2 fair coins and want to find the expected number of heads. 2. **Define the random variable:** Let $X$ be the number of heads obtained in the toss

1. **State the problem:** You draw a single card from a standard 52-card deck. If the card is red, you win 20. Otherwise, you win 0. 2. **Formula for expected value:** The expected

1. **State the problem:** You draw one coin from a bowl containing 13 pennies, 13 dimes, and 24 quarters. We want to find the expected value of the coin drawn. 2. **Formula for exp

1. **State the problem:** You draw one bill from a hat containing bills of values $5$, $10$, $20$, and $100$. Each bill is equally likely to be drawn. We want to find the expected

1. **State the problem:** We are given a random variable $X$ with values $x_i = -3, 0, 4$ and corresponding probabilities $p_i = 0.4, 0.4, 0.2$. We need to find the expected value

1. **State the problem:** We have two urns, each equally likely to be chosen. Urn 1 contains 1 white and 6 red balls, Urn 2 contains 2 white and 5 red balls. A ball is drawn and it

1. **State the problem:** We need to find the probability of event $C$, denoted as $P(C)$, using the given tree diagram. 2. **Understand the tree diagram:**

1. **State the problem:** We need to find the conditional probability $P(W|C)$, which is the probability of event $W$ given that event $C$ has occurred. 2. **Recall Bayes' formula:

1. **State the problem:** We need to find the conditional probability $P(M|B)$, which is the probability of event $M$ given event $B$. 2. **Recall the formula for conditional proba

1. The problem asks to find the probability $P(B)$ using the tree diagram. 2. According to the problem, $P(B) = P(M \cap B) + P(N \cap B)$.

1. The problem asks to find the probability $P(M \cap B)$ using the given tree diagram. 2. According to the multiplication rule for probabilities in a tree diagram, the probability

1. **State the problem:** Jaime rolls a number cube (with faces 1 to 6) and spins a spinner divided into three equal sections: Red, Blue, and Green. We want to understand all possi

1. **Stating the problem:** We have a spinner divided into six equal sections labeled 4, 1, 1, 2, 3, 1. Seven players spin the spinner and their outcomes are recorded. 2. **Underst