We conduct an experiment of counting the number of tosses of a fair coin until getting three consecutive heads. We record the flips until we get three heads (for example,H T T T T H H H). Let's define a geometric random variableXthat represents the number of tosses of a fair coin until we get three consecutive heads. Possible values forXare positive integers, such asx {1,2,3,4...}.

1. (a)Enumerate all possible outcomes with length 3, 4, 5, or 6.
2. (b)These records give us a sample space, and we can decomposeinto four sets:
3. A1={(H,H,H)}
4. A2={(T ,,, ...,)}(e.g., all records starting withT)
5. A3={(H,T,,,...,)}(e.g., all records starting withH,T)
6. A4={(H,H,T,,,...,)}(e.g., all records starting withH,H,T)
7. Please computeP(A1),P(A2),P(A3), andP(A4).
8. (E.g.,P(A2)is the probability that a randomly chosen record is in the setA2.)

(c) Please findE[X]using these four decompositions.

Hint: You could give me equation usingE[X],E[X+ 1],E[X+ 2], andE[X+ 3].