Question

1. be recorded. Define events Eli heads are observed, of the two faces is equally likely to hand face-up. Three balanced coins will be flipped once and the number of heads that land face-up is to 0,1,2,3. Note: "balanced" means each a. List all the outcomes in the Sample Space. b. Calculate the probability of each event, Eo,.., E c. Let A be the event "at least one head is observed". Calculate P(E2A) and P(AIE2)

Answer 1

$a.~Sample~space=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}\\ where~H~stands~for~head~and~T~stands~for~tail.\\ b.~P(E_0)=P(\{TTT\})=\frac{1}{2^3}=\frac{1}{8}\\ P(E_1)=P(\{HTT,THT,TTH\})=\frac{3}{2^3}=\frac{3}{8}\\ P(E_2)=P(\{HHT,THH,HTH\})=\frac{3}{2^3}=\frac{3}{8}\\ P(E_3)=P(\{HHH\})=\frac{1}{2^3}=\frac{1}{8}\\ c.~P(A)=P(\{HHH,HHT,HTH,HTT,THH,THT,TTH\})=\frac{7}{8}\\ P(E_2|A)=\frac{P(A\cap E_2)}{P(A)}=\frac{P(E_2)}{P(A)}=\frac{3/8}{7/8}=\frac{3}{7}\\ P(A|E_2)=\frac{P(A\cap E_2)}{P(E_2)}=\frac{P(E_2)}{P(E_2)}=1$