Question

7. Suppose that after the 2020 presidential election there are three races that are too close to call: Wisconsin, Michigan, and Pennsylvania. In order to become president, a candidate must receive 270 electoral votes or more. Wisconsin has 10 electoral votes, Michigan has 16, and Pennsylvania has 20. Suppose that in the other decided races, Democrats have 259 electoral votes and Republicans have 233. The Democrat has an 80% chance of winning Wisconsin, a 50% chance of winning Michigan, and a 10% chance of winning Pennsylvania. A statistician defines a random variable X to be +1 if the Democrat wins, -1 if the Republican wins, and 0 if neither candidate reaches 270 electoral votes. Find the probability density function of X. (Hint: Make this easier by figuring out the scenario in which the Republican wins or there is a tie!)

Answer 1

4)

P(0 heads) = P(fair coin)*P(both tails) + P(weighted coin)*P(both tails)=1/3*1/2*1/2+2/3*1/4*1/4=0.125

P(1 head)=P(fair coin)*P(one tail and one head) + P(weighted coin)*P(one tail and one head)=1/3*1/2*1/2*2+2/3*1/4*3/4*2=0.416666667

P(2 heads)= P(fair coin)*P(both heads) + P(weighted coin)*P(both heads)=1/3*1/2*1/2+2/3*3/4*3/4=0.458333333

PDF is

X	P(X)
0	0.125
1	0.416667
2	0.458333

5)

P(same numbers) = 1/1000*1/1000= 1/1000000= 0.000001

X (amount gain)	P(X)
90	0.000001
-10	0.999999