Q: Probability mass function statistics jersey question Three New England Patriots fans attend a Patriots-Broncos game in the stadium. The probability that the first fan will wear their "Patriots" jersey is 0.83. The probability that the second fan will wear their "Patriots" jersey is 0.51. The probability that the third fan will not wear their "Patriots" jersey is 0.55. Let X be a random variable which measures how many of the three Patriots fans mentioned are wearing their "Patriots" jersey to this football game.
Assuming that each "Patriots" fan mentioned wears their "Patriots" jersey independently of each other, find the probability distribution of X. 

P(X=0)= 0
P(X=1)= 0.83
P(X=2)= 1.02
P(X=3)= 1.65

I got X=0 since 0 Patriot fans will wear their jersey so it's a 0 percent chance
Then 
X = 1 is 1 * 0.83
X = 2 is 2 * 0.51
X = 3 is 3 * 0.45
What are the correct solutions and answers?

A: Hint: Let $A,B,C$ represent the events that the first, second, and third fan, respectively, wear their "Patriots" jersey.   You have been provided the probabilities for these events, (well, for  complement in the case of the third), and been instructed to assume independence.
Then $$\def\P{\mathop{\sf P}}\begin{align} \P(X=0) ~&=~ \P(A^\complement)\P(B^\complement)\P(C^\complement) \\[2ex]\P(X=1) ~&=~ \P(A)\P(B^\complement)\P(C^\complement)+\P(A^\complement)\P(B)\P(C^\complement)+\P(A^\complement)\P(B^\complement)\P(C)\\[1ex] &=~ \P(A)+\P(A^\complement)\,\big(\P(B)+\P(B^\complement)\P(C)\big)\end{align}$$
And so on.
