[SOLVED] Math154 homework 1 p0

MATH154
Homework 1

Problem 1.2: The card game ”set” contains 81 = 34 cards. Each card has one of 3 colors, one of 3 numbers, one of 3 shapes and one of 3 shades. It so models so the 4-dimensional vector space which is also called the field GF(81). A collection of three cards is called a ”set”, if in each of the 3
categories, all three properties either agree or are all different. You randomly pick 3 cards from the 81. What is the probability to draw a set?
Probability
Problem 1.1: a) You pick a random point (x,y) in the square [−1,1]× [−1,1]. What is the probability that x2 + y2 ≤ 1?
b) You pick a random point (x,y,z) in the unit cube [−1,1]3. What is the probability that x2 + y2 + z2 ≤ 1?
c) What is the probability that 1 if the point x
= (x1,…,x1000) is chosen randomly in the 1000-dimensional unit cube [−1,1]1000.

Figure 1. What is the probability to hit the sphere?
Probability Theory

Figure 2. The game of set visualizes a 4 dimensional vector space
Problem 1.3: The probability density of a positive integer smaller than n is prime is about 1/log(n) by the prime number theorem. What do you expect is the expected number of prime twins smaller than n?
Problem 1.4: a) Alex has three kids, and one of them is a girl. What is the probability that Alex has three girls?
b) Alex has three kids of different age and the oldest is a girl. What is the probability that Alex has three girls?
Problem 1.5: There are three boxes: a box containing two gold coins, a box containing two silver coins, and a box containing one gold coin and one silver coin. The three boxes are shuffled. You pick one box and pick a random coin from it. You notice it to be gold. What is the probability that the other coin from the same box is gold?