Probability Calculus 2020/2021, Homework 3 (two problems)
Name and Surname ... Student’s number ...
In the problems below, please use the following: as k – the sum of digits in your student’s number; as m – the sum of the two largest digits in your student’s number;
and as n – the smallest digit in your student’s number plus 1. For example, if an index number is 609999: k = 42, m = 18, n = 1.
Please write down the solutions (transformations, substitutions etc.), and additio- nally provide the final answer in the space specified (the answer should be a number in decimal notation, rounded to four digits).
3. There are n+1 white balls and k(n+1) black balls in a box. One of the white balls and several black balls were marked with a red dot (the other balls do not have marks). We randomly draw a ball from the box and we consider the following events: A = { the drawn ball is white }, B = { the drawn ball wasn’t marked with a red dot }. We know that events A and B are independent. How many black balls were marked with a red dot?
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4. An IT specialist supervises the work of 2n servers, each of which supports m2 password- protected programs. Each day, the IT specialist randomly chooses a server and one of the programs it supports (each choice has the same probability, choices on different days are independent), and changes the password for this program. Using the Poisson theorem, approximate the probability that during 2kmn subsequent days, the IT specialist will change the password for the shortest program at least twice.
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