🎲 probability

1. **State the problem:** We have a random variable $X$ uniformly distributed on the interval $(-3,5)$, and another random variable $Y = e^{-X/3}$. We want to find the expected val

1. **State the problem:** We want to find the probability that Bob plays football on exactly 2 out of the next 3 Saturdays. 2. **Identify the type of problem:** This is a binomial

1. The problem asks for the probability of picking a joker from a standard deck of cards. 2. A standard deck of cards typically contains 52 cards plus 2 jokers, making a total of 5

1. The problem asks for the probability of picking a joker from a standard deck of cards. 2. A standard deck of cards typically contains 52 cards plus 2 jokers, making a total of 5

1. **State the problem:** There are 50 students, and 15 are chosen at random. We want the probability that you or your friend (or both) are chosen. 2. **Formula and rules:** The to

1. **Problem statement:** There are 50 students in a class, and the professor chooses 15 students at random. We want to find the probability that you or your friend (or both) are a

1. **Problem Statement:** We need to formulate a probability mass function (PMF) for the number of days Ingrid takes the yellow bus to school over three consecutive days. 2. **Unde

1. The problem is to evaluate the function $f(x) = \frac{\binom{4}{x}}{16}$ for $x = 0, 1, 2, 3, 4$ and understand its values. 2. The binomial coefficient $\binom{n}{k}$ represents

1. **Problem Statement:** We have two unbiased dice with random variables $X$ and $Y$ representing the numbers on each die. We want to find the probability mass functions (PMFs) of

1. **State the problem:** We want to find the probability of getting no prime numbers when two dice are thrown. 2. **Identify prime numbers on a die:** The numbers on a die are 1,

1. **Stating the problem:** We are given that the probability it will rain on any day is $\frac{1}{6}$. We need to find:

1. **Problem Statement:** We have a uniform distribution for the average number of donuts eaten by a nine-year-old child per month, ranging from 1 to 6 donuts inclusive.

1. **Problem Statement:** A random variable $X$ has a probability distribution given by

1. **Problem Statement:** Calculate the marginal probabilities from the joint probability table for events $A_1, A_2, A_3$ and $B_1, B_2$. 2. **Given Joint Probabilities:**

1. The problem asks to fill in a 3x4 table showing the outcome, probability, and payout. 2. To solve this, we need to understand what the outcomes are, their probabilities, and the

1. **Problem Statement:** You pay 1 to play a game where two fair dice are rolled. You win 3 if the sum is 6, 7, or 8; you win 5 if the sum is 2 or 12; otherwise, you lose your 1.

1. **State the problem:** We have a probability distribution for items sold by the Booster Club with some probabilities given and one missing. We need to find the missing probabili

1. **State the problem:** We have a discrete random variable $X$ representing the number of patients visiting a clinic each day with probabilities: $$P(0) = 0.2, \quad P(1) = 0.3,

1. **State the problem:** We need to find the probability that Miss Robinson does not win the raffle. 2. **Given information:**

1. **Stating the problem:** We have a bag with counters of four colors: pink, yellow, green, and blue. The probabilities of drawing each color are given as pink 0.5, yellow 0.2, gr

1. **Stating the problem:** We have a bag containing counters of four colors: pink, yellow, green, and blue. We want to understand the probabilities of randomly drawing each color