3. Consider a modified version of the PLolya urn model in Problem 2 where we put the observed ball

back into the urn together with a new ball of the other color. For instance, suppose we randomly

sample a ball from the urn and observe that it is red. We now put the red ball back to the urn

together with a new blue ball.

(a) Write an R function PolyaUrn v2(n.steps, urn = c("red","blue")) that works for this new

sampling scheme. The input and output of the function should be the same as in Problem 2.

Hints: When updating urn in the for loop, you should probably consider using conditional

expressions.

(b) Run PolyaUrn v2(n.steps = 200) once and plot the sample path of p.red. In the plot, make

sure to

set the y-axis limit to c(0,1);

properly label the axes;

add a horizontal reference line at 0.5 (abline(h = 0.5)).

(c) Repeat PolyaUrn v2(n.steps = 200) 300 times and graph all sample paths in a single plot.

Briefly comment on the difference between this graph and the one in 2(c).