Math- 338: Solutions to Lab2 Part 2
Solution to Problem 2
The gist of this problem is to show that as the number of simulations increases, the relative
frequency of simulated outcomes in each experiment converge to their expected values under
both fair and unfair scenarios. Here, I have tabulated the results of my simulation for the
fair die. Please note that due its random nature, there is no reason that your simulation
values should exactly match mine. However, your simulations should show that the relative
frequency of each outcome in the die experiment converge to 1
6=.1666.
a1=round(runif(10,0.5,6.5))
a2=round(runif(100,0.5,6.5))
a3=round(runif(1000,0.5,6.5))
a4=round(runif(10000,0.5,6.5))
a5=round(runif(100000,0.5,6.5))
a6=round(runif(1000000,0.5,6.5))
number of simulations 1 2 3 4 5 6
10 0.2 0.1 0.2 0.1 0.3 0.1
100 0.17 0.16 0.18 0.21 0.16 0.12
1000 0.165 0.147 0.174 0.170 0.186 0.158
10000 0.1710 0.1628 0.1680 0.1679 0.1643 0.1660
100000 0.16452 0.16668 0.16742 0.16766 0.16606 0.16766
1000000 0.167065 0.167098 0.166617 0.167093 0.166054 0.166073
Table 1: Relative frequency of each outcome for varying number of simulations.
Suppose now, that the aim is to simulate rolling of an unfair die, the one with probabil-
ities: P(1) = .2, P (2) = .1, P (3) = .1, P (4) = .2, P (5) = .2, P (6) = .2. As suggested in class,
we may use the command sample.
Here is a simple way of simulating let’s say, 1000,000 rolls of the unfair die:
> a=sample(c(1,2,3,4,5,6),1000000,replace=T,prob=c(.2,.1,.1,.2,.2,.2))
> length(a[a==1])/1000000
[1] 0.1998
1
