🎲 probability

1. **Problem 1:** Subway station gates problem. (a) Define the sample space:

1. We are given the probability expression $$p(m / a) = \frac{p(a / m) \times p(m)}{p(a)} = \frac{0.112}{0.152} \approx 0.7368$$. 2. This is an application of Bayes' Theorem, which

1. The problem states: There are 100 applicants with 56 males (m) and 44 females (f). Given probabilities:

1. **Problem statement:** Given the joint density function $$f(x,y) = k(2x + y), \quad 2 < x < 6,\ 0 < y < 5$$

1. **Stating the problem:** You want to learn about classical probability, starting from the basics and moving to more advanced concepts.

1. **Stating the problem:** We want to find how many different sequences of heads (H) and tails (T) are possible when a coin is tossed 4 times. 2. **Understanding the situation:**

1. The problem: A coin is tossed four times, and the result (heads or tails) of each toss is recorded. We need to find how many different sequences of heads and tails are possible.

1. Let's start by stating the problem: We want to understand the expected value and variance of a binomial distribution. 2. A binomial distribution models the number of successes i

1. **State the problem:** We want to find the expected value and variance of a binomial distribution. 2. **What is a binomial distribution?** It describes the number of successes i

1. The problem states that autos arrive at a tollbooth at a rate of 50 autos per minute from 12:00 to 13:00. 2. We want to find the probability that the next auto will arrive withi

1. Таны асууж буй асуудал бол хоёр өөр төрлийн мөнгийг зэрэг орхих үед тодорхой үр дүн гарч магадлал ямар байх талаар юм. 2. Нэг шоо, нэг 30 талтай мөнгө орхингуут шоо дээр "3" гэс

1. Problem: Find the probability of getting an even number when a die is rolled. The possible even numbers on a die are 2, 4, and 6. So, there are 3 favorable outcomes.

1. **Problem 4.1:** Two fair dice are rolled. Find the probability that the sum of the scores is a prime number. 2. Each die has outcomes from 1 to 6, so there are $6 \times 6 = 36

1. **Stating the problem:** We have a country where 80% of residents live in the city and 20% live outside. We know that among residents without a tractor license, 13/16 live in th

1. প্রথমে সমস্যাটি বুঝি: দুটি বাক্স থেকে একটি করে বল নেওয়া হয়েছে, প্রথম বাক্সে ৩টি সাদা এবং ৫টি লাল বল, দ্বিতীয় বাক্সে ৪টি সাদা এবং ৯টি লাল বল আছে। 2. ঘটনাগুলো সংজ্ঞায়িত করি:

1. সমস্যা: দুটি বাক্স থেকে একটিতে সমসময়ের ভিত্তিতে একটি বাক্স নির্বাচন করা হয়। প্রথম বাক্সে ৩টি সাদা এবং ৫টি লাল বল, দ্বিতীয় বাক্সে ৪টি সাদা এবং ৯টি লাল বল রয়েছে। যে বলটি বেছে

1. **Problem statement:** The random vector $(X,Y)$ is uniformly distributed over region $A$ defined by vertices $(1,1)$, $(1,0)$, $(2,0)$, and $(2,2)$. 2. **Describe region $A$ al

1. **Understanding the problem:** We are given the probabilities of two players, Encik Wilson and Encik Alvin, making a "hole in one" to hole S. The probability that Encik Wilson m