University of Victoria MATH 352 A1_Soln_Spring_2019

University of VictoriaMATH 352A1_Soln_Spring_2019 Math 352, Introduction to probability: Assignment 1 Due date: 23rd January, 2019 1. Consider the experiment: We toss a coin until we obtain a head". Write down the sample space for this experiment. Write down the event: E = fWe obtain a head within 3 tossesg as a subset of the sample space. Solution. Sample space Ω = ffHg; fT Hg; fTT Hg; fTTT Hg; : : :g. Event E = ffHg; fT Hg; fTT Hgg. 2. We take a rod of length 1 metre and randomly divide it into two pieces. Write down the sample space for this experiment. Can you plot it? Write down the event E that at least one of the pieces has length strictly bigger than 1=2 metre. Can you plot the sample space as a subset of the plane R2? Can you describe the event E as a subset of R2? Solution. The sample space is Ω := f(x; y) : x ≥ 0; y ≥ 0; x + y = 1g. The event is E = Ω n f(1=2; 1=2)g since the only way one of the pieces is not strictly bigger than 1=2 is if both the pieces have length equal to 1=2. The plot of Ω is given below. E is the same plot without the point (1=2; 1=2). (1; 0) (0; 1) 3. How many ways are there to color n different balls with 4 colors? If one of the above colorings are chosen uniformly at random, what is the probability that at least two balls will have different colors?