# Συνδυαστική/Πιθανότητες

How Many Friends You Need to Have a Birthday Every Day of the Year. It seems like every day I log in to Facebook, it’s someone’s birthday, and I’m not even a very active user.

I think it’s been at least a year since I last friended someone. Maybe two. Also, I have fewer than 365 friends on the social network, so it’s mathematically impossible for there to be a birthday every single day of the year. But still, it seems like a lot of days with at least one birthday. Two questions: How many friends do you need to nearly guarantee that during any given day of the year, it’s at least one friend’s birthday? I was sure someone had answered these question before, so I googled it first. More generally, in probability theory, this is known as the coupon collector’s problem. Here’s what the probabilities look like, as it pertains to our birthday problem. So assuming birthdays are random and disregarding leap birthdays, you have about a 50-50 chance that any given day of the year is some friend’s birthday with 2,287 friends or more.

That answers the first question. The Loaded Dice Puzzle. And My Reaction to Power-Up, A New Book – Mind Your Decisions. A standard dice shows the number 6, and every number, with probability 1/6.

Decisionsciencenews. Main TWO SHOT RUSSIAN ROULETTE WITH 2 BULLETS OR AN EQUAL CHANCE OF 1 or 3 BULLETS With p as the probability of dying on one shot, this figure shows how to get the probability of living through the game.

A recipe for dynamic programming in R: Solving a “quadruel” case from PuzzlOR – Serhat's Blog. As also described in Cormen, et al (2009) p. 65, in algorithm design, divide-and-conquer paradigm incorporates a recursive approach in which the main problem is: Divided into smaller sub-problems (divide),The sub-problems are solved (conquer),And the solutions to sub-problems are combined to solve the original and “bigger” problem (combine).

Instead of constructing indefinite number of nested loops destroying the readability of the code and the performance of execution, the “recursive” way utilizes just one block of code which calls itself (hence the term “recursive”) for the smaller problem. The main point is to define a “stop” rule, so that the function does not sink into an infinite recursion depth. Book Proofs – A blog for mathematical riddles, puzzles, and elegant proofs. This Riddler puzzle is about a card game where the goal is to find the largest card.

Approximate solution Let’s suppose that the deck has nn cards in it, and we are dealt kk cards from the deck. Every time a card is flipped, we must decide whether to keep playing or to stop. Suppose we have flipped over mm cards already and the largest card flipped so far has a value of aa. Can You Solve The Probability Two Random Walkers Meet? A Strange Way To Approximate Pi. Sunday Puzzle – Mind Your Decisions. Points A and B are opposite corners of a 5×5 grid.

Alice starts at point A, and each second, walks one edge right or up (if a point has two options, each direction has a 50% chance) until Alice reaches point B. At the same time, Bob starts at point B, and each second he walks one edge left or down (if a point has two options, each direction has a 50% chance) in order to reach point A. 40 Questions on Probability for data science - [Solution: SkillPower - Probability, DataFest 2017] Introduction Probability forms the backbone of many important data science concepts from inferential statistics to Bayesian networks.

It would not be wrong to say that the journey of mastering statistics begins with probability. This skilltest was conducted to help you identify your skill level in probability. A total of 1249 people registered for this skill test. The test was designed to test the conceptual knowledge of probability. What Is The Probability It Will Rain Forever? Sunday Puzzle – Mind Your Decisions. Weather in Mathland is determined by chance.

-If it rains on a given day, then the probability it will rain the next day increases by 10 percentage points (up to 100%). -If it does not rain on a given day, then the probability it will rain the next day decreases by 10 percentage points (down to 0%). If the probability it will rain today is 60%, what is the probability that it will eventually rain every day, for all time? Watch the video for a solution. Can You Solve The Probability It Will Rain Forever Puzzle? Or keep reading. The Missing Sock Problem – Sunday Puzzle – Mind Your Decisions. A statistician keeps a simple wardrobe.

He only purchases pairs of black socks and white socks, and he keeps all of the socks in a pile in the drawer. Recently one of the socks was lost in the laundry, resulting in a mathematical property. If you select two socks at random from the drawer, the socks will match in color exactly 50% of the time. The statistician owns more than 200 socks but less than 250 socks, and there are more black socks than white socks.

Can You Solve The Probability The Newborn Is A Boy? Technical Interview Question – Sunday Puzzle – Mind Your Decisions. On the morning of January 1, a hospital nursery has 3 boys and some number of girls.

That night, a woman gives birth to a child, and the child is placed in the nursery. On January 2, a statistician conducts a survey and selects a child at random from the nursery (including the newborn and every child from January 1). The child is a boy. What is the probability the child born on January 1 was a boy? This problem has been asked as a technical interview question. Can You Solve Google’s Car Probability Interview Question? Sunday Puzzle – Mind Your Decisions. Here is a problem that Google asked as an interview question. If the probability of seeing a car on the highway in 30 minutes is 0.95, what is the probability of seeing a car on the highway in 10 minutes? (assume a constant default probability) Some clarifications that can help are:

The Race To December 31 – Sunday Puzzle – Mind Your Decisions. Here’s a fun game you can play with another person. The game starts on January 1. Each of two players takes turns calling out another date. The new date has to be a later date in the year with either the same month OR the same day (from January 1, a player can call out a later day in January or another month with the day 1 like February 1, March 1, etc.). The person who calls out December 31 wins the game.

Who has the winning strategy, the first or second player? This game was covered by Grey Matters in 2010, who communicated it to ScamSchool who made a video. Διασκεδαστικά Μαθηματικά: Κοινοβουλευτική πρακτική. Διασκεδαστικά Μαθηματικά: Λύτες προβλημάτων. Distance Between Two Random Points In A Square – Sunday Puzzle – Mind Your Decisions.

What is the average distance between two randomly chosen points in a square with a side length 1? Specifically, select two points at random (drawn from the standard uniform distribution) from the interior of the unit square. What is the mean distance between the two random points? *Warning: finding the exact answer involves tricky integrals. Watch the video for a solution. Μισθολογικές ανισότητες. Michael Fenton’s counting problem. Another riddle. A very nice puzzle on The Riddler last week that kept me busy on train and plane rides, runs and even in between over the weekend. The core of the puzzle is about finding the optimal procedure to select k guesses about the value of a uniformly random integer x in {a,a+1,…,b}, given that each guess y produces the position of x respective to y (less, equal, or more).

If y=x at one stage, the player wins x. Optimal being defined as maximising the expected gain. After some (and more) experimentation, I found that, when b-a is large enough [depending on k], the optimal guess at stage i is b-f(i) with f(k)=0 and f(i-1)=2f(i)+1. For the values given on The Riddler, a=1,b=1000,k=9, my solution is to first guess at y=1000-f(9)=255 and this produces a gain of 380.31 with a probability of winning of 0.510, which seems amazingly large, but not so much when considering that 2⁹ is close to 500. for small enough values of b-a. Διασκεδαστικά Μαθηματικά: Το δείπνο του κ. Καθηγητή ( Rolling Dice ( To continue to be a valuable Olympiad resource, AoPS can use your help adding to Contest Collections. How to post problems: In the High School Olympiads forum, use the New Topic button in the top right to create a new topic. When Warren Buffett Challenged Bill Gates To A Dice Game – Sunday Puzzle – Mind Your Decisions.

Photo by Ella’s Dad, CC BY 2.0 Warren Buffett once challenged Bill Gates to a game of dice similar to this. Each would pick a die, they would roll many times, and the person who rolled the higher number more often would win. The four dice all had 6 faces equally likely to show, but their faces had the following numbers. Number Of Ways To Divide Coins Interview Question – Sunday Puzzle – Mind Your Decisions. Combinatorics Problem on Geometric Distribution: Can You Roll The Die? - Soumik Chakraborty. Διασκεδαστικά Μαθηματικά: Αναλογικό ρολόι. Odd Numbers in Pascal's Triangle.

Pascal's Triangle has many surprising patterns and properties. For instance, we can ask: "how many odd numbers are in row N of Pascal's Triangle? " For rows 0, 1, ..., 20, we count: row N: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 odd #s: 1 2 2 4 2 4 4 8 2 4 04 08 04 08 08 16 02 04 04 08 04. Secret Santa Surprising Probability – Sunday Puzzle. A group of people decide to do a Secret Santa. A hat contains slips of paper with each person’s name. Οι Τρόποι.

Library(e1071) library(dplyr) c("A1","A2","A3","C1","C2") %>% length %>% permutations %>% data.frame %>% mutate_each(funs(x[.])) %>% filter (!X1 %in% c("A3","C1","C2")) %>% filter (!X2 %in% c("C1","C2")) – gd047

How Many 3×3 Magic Squares Are There? Sunday Puzzle. A magic square is a 3×3 grid where every row, column, and diagonal sum to the same number. Dice rolls until a repeat – Sunday Puzzle. Διασκεδαστικά Μαθηματικά: Διαδρομές. Διασκεδαστικά Μαθηματικά: 5 πριν το 7. Βάτραχος vs ...μύγες. Πιθανότητες και Ιατρική. Ψαρομεζές. 7 ντόμινο. Νο \$123456789\$ Πεντάγωνο νησί. To Pizza or not to Pizza? Solutions to knight's random walk. Πρώτο ποιο; Ουρά...... Δυο διαγωνιστικά προβληματάκια ιδίου φυράματος για παιδιά γυμνάσιου, το πρώτο ελαφρά παραλλαγμένο από υλικό μαθημάτων προετοιμασίας της μαθηματικής εταιρείας της Νοτίου Αφρικής και το άλλο -πολύ γνωστό-από ένα Θαλή του 97. 1) Όταν ο Παπαδόπουλος πήγε στην εφορία για να τακτοποιήσει κάποιες εκκρεμότητες είδε ότι υπήρχε μόνο ένα γκισέ και μια μεγάλη ουρά. Κάθισε λοιπόν στο τέλος της ουράς και περίμενε. 8 - ψήφιοι αριθμοί. Χρόνος ζωής. Ασκήσεις ιστορικού ενδιαφέροντος - Πιθανότητες (6)

Τρία κουτιά. Το πρόβληµα του σπιρτόκουτου του Banach. ▪ Τεσσάρι στο ΛΟΤΤΟ. Στο τυχερό παιχνίδι του ΛΟΤΤΟ "6 από 49", αν παίξουμε μια στήλη, ποια είναι η πιθανότητα του ενδεχομένου Α: "να πετύχουμε 4 ακριβώς σωστά νούμερα"; Επειδή τελικά δεν έχει σημασία η σειρά κλήρωσης του κάθε αριθμού, οι δυνατές περιπτώσεις του πειράματος είναι τόσες όσοι και οι συνδυασμοί των 49 ανά 6, δηλαδή Για να βρούμε το πλήθος των ευνοϊκών περιπτώσεων σκεφτόμαστε ως εξής: Υπάρχουν \$\binom{6}{4}\$ τρόποι για να επιλέξουμε 4 σωστά νούμερα από τα 6 που κληρώθηκαν. Στη συνέχεια μένουν \binom{49-6}{6-4}=\binom{43}{2} τρόποι για να επιλέξουμε τα 2 λάθος νούμερα.

▪ 365 άτομα. 5 για 2 και 2 για 5.