A coin is tossed three times, where

(i) $E$ : head on third toss, $F$ : heads on first two tosses

(ii) $E$ : at least two heads, $F$ : at most two heads

(iii) $E$ : at most two tails, $F$ : at least one tail

.: When a coin is tossed three times, the sample space $S$ contain 8 equally likely sample points.

$\Rightarrow S = \{ HHH,\;HHT,\;HTH,\;HTT,\;THH,\;THT,\;TTH,\;TTT\}$

(i) Let $E$ : ‘head on third toss’ and $F$ : ‘heads on first two tosses’

$\Rightarrow E = \{ HHH,\;HTH,\;THH,\;TTH\}$ and $F = \{ HHH,\;HHT\}$

$F = \{ HHH,\;HHT\} \Rightarrow E \cap F = \{ HHH\}$

$P(E) = \cfrac{4}{8} = \cfrac{1}{2},P(F) = \cfrac{2}{8} = \cfrac{1}{4}$

and $P(E \cap F) = \cfrac{1}{8}$

$\therefore$ $P(E|F) = \cfrac{{P(E \cap F)}}{{P(F)}} = \cfrac{{1/8}}{{2/8}} = \cfrac{1}{2}.$

(ii) Let $E$ : ‘atleast two heads’ and $F$ : ‘atmost two heads’ $\Rightarrow E = \{ HHH,\;HHT,\;HTH,\;THH\}$

and
$F = \{ TTT,\;THT,\;TTH,\;HTT,\;HHT,\;HTH,\;THH\}$

$\Rightarrow E \cap F = \{ HHT,\;HTH,\;THH\}$

Hence, $P(E) = \cfrac{4}{8} = \cfrac{1}{2},P(F) = \cfrac{7}{8}$

and $P(E \cap F) = \cfrac{3}{8}$

$P(E|F) = \cfrac{{P(E \cap F)}}{{P(F)}} = \cfrac{{3/8}}{{7/8}} = \cfrac{3}{7}$

(iii) Let $E$ : ‘atmost two tails’ and $F$ : ‘atleast one tail’.

$\Rightarrow E = \{ HHH,\;HHT,\;HTH,\;HTT,\;THH,\;THT,\;TTH\}$

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

$\Rightarrow E \cap F = \{ HHT,\;HTH,\;HTT,\;THH,\;THT,\;TTH\}$

Hence, $P(E) = \cfrac{7}{8},P(F) = \cfrac{7}{8}$

and $P(E \cap F) = \cfrac{6}{8}$

$\therefore$ $P(E|F) = \cfrac{{P(E \cap F)}}{{P(F)}} = \cfrac{{6/8}}{{7/8}} = \cfrac{6}{7}$