Question

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

Answer 1

It is given that a coin is tossed three times.

The sample space S when a coin is tossed three times is,

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

The total number of sample space is 8.

(i)

Let E be the event of occurring head on third toss and F be the event of occurring heads on first two tosses.

E={ HHH,HTH,HTH,THH,TTH } P( E )= 5 8

And,

F={ HHH,HHT } P( F )= 2 8 = 1 4

The common occurring between events E and F is,

E∩F={ HHH } P( E∩F )= 1 8

The probability P( E|F ) is calculated as,

P( E|F )= P( E∩F ) P( F ) = 1 8 1 4 = 4 8 = 1 2

Therefore, the value of P( E|F ) is 1 2 .

(ii)

Let E be the event of occurring at least two heads and F be the event of occurring at most two heads.

E={ HHH,HHT,HTH,THH } P( E )= 4 8 = 1 2

And,

F={ HHT,HTH,HTT,THH,TTH,THT,TTT } P( F )= 7 8

The common occurring between events E and F is,

E∩F={ HHT,HTH,THH } P( E∩F )= 3 8

The probability P( E|F ) is calculated as,

P( E|F )= P( E∩F ) P( F ) = 3 8 7 8 = 3 7

Therefore, the value of P( E|F ) is 3 7 .

(iii)

Let E be the event of occurring at most two tails and F be the event of occurring at least one tail.

E={ HHH,HHT,HTT,HTH,THH,THT,TTH } P( E )= 7 8

And,

F={ HHT,HTT,HTH,THH,THT,TTH,TTT } P( F )= 7 8

The common occurring between events E and F is,

E∩F={ HHT,HTT,HTH,THH,THT,TTH } P( E∩F )= 6 8 = 3 4

The probability P( E|F ) is calculated as,

P( E|F )= P( E∩F ) P( F ) = 3 4 7 8 = 6 7

Therefore, the value of P( E|F ) is 6 7 .