In each of 4 races, the Democrats have 60% chances of winning. Assuming that the races are independent of each other, find the probability that: (1) the Democrats will win 0 races, 1 race, 2 race, 3 race, or all 4 races? (2) the Democrats will win a majority of the races.

Let $X=$ the number of wins of the Democrats out of 4 races:

X$\sim Bin(n,p)$

Given $n=4, p=0.60, q=1-p=0.40.$

(1)

$P(X=0)=\dbinom{4}{0}(0.6)^0(0.4)^{4-0}=0.0256$$P(X=1)=\dbinom{4}{1}(0.6)^1(0.4)^{4-1}=0.1536$$P(X=2)=\dbinom{4}{2}(0.6)^2(0.4)^{4-2}=0.3456$$P(X=3)=\dbinom{4}{3}(0.6)^3(0.4)^{4-3}=0.3456$$P(X=4)=\dbinom{4}{4}(0.6)^4(0.4)^{4-4}=0.1296$

(2)

$P(X>2)=P(X=3)+P(X=4)$$=0.3456+0.1296=0.4752$