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Math 221: Simulations/Law of Large Numbers The Birthday Problem Let A be the event that at least two people from a class of 50 share the same birthday. We can use simulations to find the probability of A. The exact probability is: P (A) = 1 − P (Ā) = 1 − 316 365 364 363 365 P50 = .97037 · · ··· =1− 365 365 365 365 36550 A simulation of a procedure is a process that behaves the same way as the procedure, so that similar results are produced. In this case, each probability of a birthday is assumed to be equally likely (which is a fair assumption). We can use a random number generator to randomly choose numbers between 1 and 365, where 1 indicates January 1st and 365 indicates December 31st. The Statistics toolbox for the program Matlab can easily randomly select the numbers. The simulation would proceed as follows: 1. Randomly select 50 numbers from 1 to 365 (pick 50 birthdays). 2. Determine if at least two of the 50 numbers match (at least two people have the same birthday). 3. Repeat the last two steps thousands of times, keeping track of how many times at least two of the 50 numbers matched, along with the number of trials. The programming code for the simulation along with a plot are below. n = 50; tot = 0; tot_vec = []; for i=1:10000 i G=sort(unidrnd(365,n,1)); H=[G;0] - [0;G]; yes = sum(H==0); if yes > 0, tot = tot + 1; end tot_vec = [ tot_vec; tot/i ]; end plot(tot_vec); page 2 Simulation of Probability that at least 2 people in 50 share the same birthday 1 Relative Frequency .99 .98 .97 .96 .95 .94 0 500 1000 2000 1500 2500 Number of Trials 3000 3500 4000