# A Coin Is Tossed Three Times, Where (A) E: Head On Third Toss,F: Heads On First Two Tosses(B) E: At Least Two Heads,F: At Most Two Heads(C) E : At Most Two Tails,F : At Least One Tail. Determine P(E∣F)

(A) E: Head On Third Toss, F: Heads On First Two Tosses

A coin is tossed three times

Therefore, S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

E = {HHH,HTH,THH,TTH}

$$P(E) = \frac{4}{8}$$ $$P(E) = \frac{1}{2}$$

F: Heads On First Two Tosses

F = {HHH,HHT}

$$P(F) = \frac{2}{8}$$ $$P(F) = \frac{1}{4}$$ $$(E\cap F) = {HHH}$$ $$P(E\cap F) = \frac{1}{8}$$

Now, we have to find P(E|F)

$$P(E|F) = \frac{P(E\cap F)}{R(F)}$$

= $$\frac{\frac{1}{8}}{\frac{1}{4}}$$

= $$\frac{1}{8} * \frac{4}{1}$$

= $$\frac{1}{2}$$

Therefore,$$P(E|F) = \frac{1}{2} = 0.50$$

A coin is tossed three times

Therefore, S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

E = {HHH,HTH,THH,TTH}

P(E) =$$\frac{4}{8}$$

P(E) =$$\frac{1}{2}$$

F = { HHT, THH, HTH, TTH, THT, HTT, TTT}

P(F) =$$\frac{7}{8}$$

E\cap F = {HHT,THH, HTH}\) $$P(E \cap F) = \frac{3}{8}$$

Now, we have to find P(E|F)

$$P(E|F) = \frac{P(E\cap F)}{P(F)}$$

=$$\frac{\frac{3}{8}}{\frac{7}{8}}$$

=$$\frac{3}{8} * \frac{7}{8}$$

=$$\frac{3}{7}$$

Therefore, P(E|F) =$$\frac{3}{7} = 0.42$$

(C) E: At Most Two Tails, F : At Least One Tail

A coin is tossed three times

Therefore, S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

E: At Most Two Tails

E = {HHH, HHT, HTH, THH, TTH, THT, HHT}

P(E) =$$\frac{7}{8}$$

F: At Least One Tail

F = { HHT, THH, HTH, TTH, THT, HTT, TTT}

P(F) =$$\frac{7}{8}$$ $$E\cap F = {HHT, HTH, THH, TTH, THT, HTT }$$ $$P(E \cap F) = \frac{6}{8}$$

Now, we have to find P(E|F)

$$P(E|F) = \frac{P(E \cap F)}{P(F)}$$ $$P(E|F) = \frac{\frac{6}{8}}{\frac{7}{8}}$$ $$P(E|F) = \frac{6}{8} * \frac{8}{7}$$ $$P(E|F) = \frac{6}{7}$$

Therefore,$$P(E|F) = \frac{6}{7} = = 0.85$$

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