🎲 probability

1. The problem is to find the value of $P(4)$ given the function $P(x) = p(1 - p)^{x-1}$. 2. This function is a geometric probability mass function where $p$ is the probability of

1. **State the problem:** We are given the probability mass function (PMF) of a geometric distribution: $$P(x) = p(1-p)^{x-1}$$ and asked to find $$P(4)$$ for a given probability $

1. **Problem statement:** We are given that 90% of emails are spam and 10% are legitimate. We want to find probabilities related to the first legitimate email found in a sequence o

1. **Problem Statement:** We are given that 90% of emails are spam and 10% are legitimate. We want to find probabilities related to the first legitimate email found at certain posi

1. **Problem statement:** A biased tetrahedral die with faces 0, 1, 2, 3 has probabilities $P(X=0)=\frac{1}{6}$, $P(X=1)=\frac{1}{6}$, $P(X=2)=\frac{1}{6}$, and $P(X=3)=\frac{1}{2}

1. **Problem statement:** Suppose you pick three cards randomly from a deck of 52 cards, and they happen to be aces. What is the probability that the next card drawn from the remai

1. **State the problem:** We want to find the probability that Xavier does not throw a 1 on his first throw but throws a 1 on his second throw with a fair six-sided die. 2. **Formu

1. **State the problem:** Aditi has 1 white sock and 9 yellow socks in a bag. She draws one sock, replaces it, then draws another sock. We need to find the probability that the two

1. **State the problem:** We want to find the probability that Kiaan takes two counters that are not white when he draws one counter, replaces it, and then draws another.

1. **State the problem:** We want to find the probability that at least two out of three randomly selected people share the same Chinese zodiac sign. 2. **Total zodiac signs:** The

1. **Stating the problem:** We have a sports club with members practicing Volleyball (V), Basketball (B), and Tennis (T). We want to find the probability that a randomly chosen mem

1. **State the problem:** We have a probability distribution for the rate of return with intervals and their probabilities. We define two events: - Event A: Rate of return will be

1. **Problem statement:** We have an average rate of 150 cars per hour passing a point. We want to find probabilities related to the number of cars passing in different time interv

1. **Stating the problem:** We are given that the probability a randomly selected U.S. adult drinks coffee on a given day is $0.56$.

1. **مشكلة السحب من الكيس:** نضع في كيس 3 كريات تحمل الرقم 1، 2 كريات تحمل الرقم 2، وكرة واحدة تحمل الرقم 3.

1. **State the problem:** We are given the generating function $$G_X(s) = \frac{s^3}{6} (2 + s^4 + 2s^8)$$ and asked to find $$\text{mean}(3X + 5) + \text{Var}(5X + 7).$$

1. **State the problem:** We have a bag with 7 red, 5 blue, and 4 green marbles, totaling $7 + 5 + 4 = 16$ marbles. We draw 3 marbles without replacement and want the probability t

1. **State the problem:** We have a bag with 5 red, 3 blue, and 6 green marbles, totaling $5 + 3 + 6 = 14$ marbles. We draw 3 marbles without replacement and want the probability t

1. **State the problem:** We have a bag with 5 red, 6 blue, and 4 green marbles, totaling $5 + 6 + 4 = 15$ marbles. 2. **What is asked:** Find the probability that all three marble

1. **State the problem:** We have a bag with 6 red, 7 blue, and 3 green marbles, totaling $6 + 7 + 3 = 16$ marbles. We want to find the probability that when drawing 3 marbles with

1. **Problem statement:** A die is rolled once. Find the probability that the outcome is: (a) not a 4