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: Head On Third Toss

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

(B) E: At Least Two Heads, F: At Most Two Heads

A coin is tossed three times

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

E: At Least Two Heads

E = {HHH,HTH,THH,TTH}

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

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

F: At Most Two Heads

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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