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


Determine P(E|F)


The sample space of the given experiment will be:

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


(i) Here, E: head on third toss


And F: heads on first two tosses


E = {HHH, HTH, THH, TTH} and F = {HHH, HHT}


E F = {HHH}


So,


Now, we know that


By definition of conditional probability,





(ii) Here, E: at least two heads


And F: at most two heads


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


E F = {HHT, HTH, THH}


So,


Now, we know that


By definition of conditional probability,




(iii) Here, E: at most two tails


And F: at least one tail


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


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


So,


Now, we know that


By definition of conditional probability,




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