Hi Faith,
a. The key here is identifying what events correspond to what x-values.
P(X=0) This is the probability of all heads, since heads corresponds to a score of 0. Probability of a head on any given toss is 0.2, so:
P(X=0) = 0.2^3 = 0.008
P(X=2) This is the probability of exactly one tail. We need to know how many outcomes in the three trials correspond to this. To determine this, draw a tree diagram or simply list out all possible outcomes. Tree diagrams are difficult to put in Wyzant, but outcomes are below:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Thus, there are three outcomes that correspond to exactly one tail. Now,you need the probability of getting two heads and exactly one tail. Probability of getting tails is 1-0.2= 0.8 So:
P= 0.2*0.2*0.8= 0.032
But we have three outcomes like this, so:
P(X=2) = 3(0.2*0.2*0.8) = 0.096
P(X=4) This is the probability of exactly two tails. Again, remember that P(Tails) = 0.8 Refer to the list of outcomes above. Again, we have three possibilities, so we'll multiply by 3.
P(X=4) = 3(0.8*0.8*0.2) = 0.384
P(X=6) This is the probability of getting all tails. Only one outcome here:
P(X=6) = 0.8^3 = 0.512
All of these should add to 1, which holds true for all discrete probability distributions and helps you check your work.
b. Use the Complement Rule aka the "One Minus Trick:"
P(X<6) = 1 - P(X=6)
P(X<6) = 1- 0.512 = 0.488
c. We know that it is impossible to get 1, 3, or 5 given the parameters of the problem. Every toss has to be associated with 2 or 0, and we cannot get to 1, 3, or 5 with those values. So, to get P(1<x<5), we add:
P(1<x<5) = P(X=2) + P(X=4)
P(1<x<5) = 0.096 + 0.384 = 0.48
I hope this helps.
