probability distribution

Question

suppose that one each of 5 races the democrats have a 60% chance of winning. assuming the races are independent of each other, let x be the number of races democrats win.

a) find the probability distribution of x

b) find the probability that democrats win at least one race

c) find the probability that democrats win at most 2 races.

Answer 1

Can look at this a a problem in binomial expansion of (x+y)^5 which is

x^5 + 5x^4y + 10x^3y^2 +10x^2y^3 +5xy^4 + y^5 with probability of win=0.6 and probability of loss=0.4

The only way they would not win at least one race would be all outcomes except losing 5 times. The probability of 5 loses is (0.4)^5=0.01 so probability of winning at least once is (1.0 - .01 =.99)

For them to win at most 2 races would be the probability of their winning either one or two races

Prob of winning one race is 5(0.6)(0.4)^4- = .0768

Prob of winning two races is 10(0.6)^2(0.4)^3 = .2304

so combined probability = .0768 + .2304 = .3072