Question

In each of the 4 races, the democrats have 60% chance of winning.

Assuming that the races are independent of each other, what is the

probability that:

a) The Democrats will win 0 races, 1 race, 2 races, 3 races, or 4

races?

b) The Democrats will win at least 1 race.

c) The Democrats will win a majority of the races.

Answer 1

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

X$\sim Bin(n,p)$

Given $n=4, p=0.60, q=1-p=0.40.$

a)

$$P(X=0)=\dbinom{4}{0}(0.6)^0(0.4)^{4-0}=0.0256$$

$$P(X=1)=\dbinom{4}{1}(0.6)^1(0.4)^{4-1}=0.1536$$

$$P(X=2)=\dbinom{4}{2}(0.6)^2(0.4)^{4-2}=0.3456$$

$$P(X=3)=\dbinom{4}{3}(0.6)^3(0.4)^{4-3}=0.3456$$

$$P(X=4)=\dbinom{4}{4}(0.6)^4(0.4)^{4-4}=0.1296$$

b)

$$P(X\geq 1)=1-P(X=0)$$

$$=1-0.0256=0.9744$$

c)

$$P(X>2)=P(X=3)+P(X=4)$$

$$=0.3456+0.1296=0.4752$$