Question

Example 5.5. We roll a fair die then toss a coin the number of times shown on the die. What is the probability of the event A that all coin tosses result in heads? One could use the state space Ω = {(1, H), (1, T), (2, H, H), (2, T, T), (2, T, H), (2, H, T), . . . }. However, the outcomes are then not all equally likely. Instead, we continue the state space is Ω {1, ,6} × {H,T)6 and the outcomes are equally likely. Then, the event of interest is A -A1 UA2 UA3 UA4UA5U A6, where

Answer 1

p(6bsenNǐng, at least ene head atton kAll due) 1-f (Alsey Nina, all tails atter s(eU, ot die) = tosses came up tail G L 2 4 6 32 64 2I 129 equal no c Aead tails agtex a die sou) det Ai = erent die corne ute exp toaesare 2 head t /2 tosses ane tails (i 2,4,6)

(3) 6 2-16 324 24 + 20 64 X6 19 96 G4X 6

P nlu heads te Ae evene that die came all tosaes are heads P (ernet heads)-P(A) = Σ P(Ai) に! 丄ト(1-p6) (I-p) P (at least 4 head) tat Bi event die come up i k all tosea auve toils cuuls に1 6 p (At lea at l head.) = 1-Y (onlf taik)